No title

Journal of Functional Analysis 257 (2009) 1–19 www.elsevier.com/locate/jfa

Hardy type inequality and application to the stability of degenerate stationary waves Shuichi Kawashima a,∗ , Kazuhiro Kurata b a Faculty of Mathematics, Kyushu University, Fukuoka 812-8581, Japan b Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Hachioji, Tokyo 192-03, Japan

Received 6 August 2008; accepted 7 April 2009

Communicated by C. Villani

Abstract This paper is concerned with the asymptotic stability of degenerate stationary waves for viscous conservation laws in the half space. It is proved that the solution converges to the corresponding degenerate stationary wave at the rate t −α/4 as t → ∞, provided that the initial perturbation is in the weighted space L2α = L2 (R+ ; (1 + x)α ) for α < αc (q) := 3 + 2/q, where q is the degeneracy exponent. This restriction on α is best possible in the sense that the corresponding linearized operator cannot be dissipative in L2α for α > αc (q). Our stability analysis is based on the space-time weighted energy method combined with a Hardy type inequality with the best possible constant. © 2009 Elsevier Inc. All rights reserved. Keywords: Viscous conservation laws; Degenerate stationary waves; Asymptotic stability; Hardy inequality

1. Introduction We study the stability problem of degenerate stationary waves for viscous conservation laws in the half space x > 0: ut + f (u)x = uxx , u(0, t) = −1,

u(x, 0) = u0 (x).

* Corresponding author.

E-mail addresses: [email protected] (S. Kawashima), [email protected] (K. Kurata). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.04.003

(1.1)

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Here the initial function is assumed to satisfy u0 (x) → 0 as x → ∞, and f (u) is a smooth function of the form f (u) =

  1 (−u)q+1 1 + g(u) , q

f  (u) > 0 for −1  u < 0,

(1.2)

where q is a positive integer (degeneracy exponent) and g(u) = O(|u|) for u → 0. Since f (0) = f  (0) = 0 and f (u) is strictly convex for −1  u < 0, we see that f (u) > 0 for −1  u < 0. In particular, we have 1 + g(u) > 0 for −1  u  0. In this situation, the corresponding stationary problem admits a unique solution φ(x) (called degenerate stationary wave), which verifies φx = f (φ), φ(0) = −1,

φ(x) → 0 as x → ∞.

(1.3)

We see easily that φ(x) behaves like φ(x) ∼ −(1 + x)−1/q . In particular, we have φ(x) = −(1 + x)−1/q when g(u) ≡ 0. To discuss the stability of the degenerate stationary wave φ(x), we introduce the perturbation v by u(x, t) = φ(x) + v(x, t) and rewrite the problem (1.1) as   vt + f (φ + v) − f (φ) x = vxx , v(0, t) = 0,

v(x, 0) = v0 (x),

(1.4)

where v0 (x) = u0 (x) − φ(x), and v0 (x) → 0 as x → ∞. The stability of degenerate stationary waves was first studied in [15]. It was proved in [15] that if the initial perturbation v0 (x) is in the weighted space L2α , then the perturbation v(x, t) decays in L2 at the rate t −α/4 as t → ∞, provided that α < α∗ (q), where    α∗ (q) := q + 1 + 3q 2 + 4q + 1 /q. The decay rate t −α/4 obtained in [15] would be optimal but the restriction α < α∗ (q) was not very sharp. The main purpose of this paper is to relax this restriction. Indeed, by employing the space–time weighted energy method in [15] and by applying a Hardy type inequality with the best possible constant (see Proposition 2.3), we show the same decay rate t −α/4 under the weaker restriction α < αc (q) := 3 + 2/q (see Theorem 4.1). Notice that α∗ (q) < αc (q). It is interesting to note that a similar restriction on the weight is imposed also for the stability of degenerate shock profiles (see [10]). We remark that our stability result for degenerate stationary waves is completely different from those for non-degenerate case. In fact, for non-degenerate stationary waves, we have the better decay rate t −α/2 for the perturbation without any restriction on α. See [4–6,14,16] for the details. See also [2,7,9,11] for the related stability results for stationary waves. In this paper we also discuss the dissipativity of the following linearized operator associated with (1.4):   Lv = vxx − f  (φ)v x .

(1.5)

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In a simpler situation including the case g(u) ≡ 0 in (1.2), we show that the operator L is uniformly dissipative in L2α for α < αc (q) but cannot be dissipative for α > αc (q) (see Theorem 3.5). This suggests that the exponent αc (q) is the critical exponent of the stability problem of degenerate stationary waves. This result on the characterization of the dissipativity of L is an improvement on the previous one in [15] and is established by using a Hardy type inequality with the best possible constant (see Proposition 2.3). The contents of this paper are as follows. In Section 2 we introduce several Hardy type inequalities and discuss the best possibility of their constants. In Section 3 we discuss the dissipativity of the operator L in (1.5) in weighted L2 spaces. Finally in Section 4, we study the nonlinear stability of degenerate stationary waves. Notations. For 1  p  ∞, Lp = Lp (R+ ) denotes the usual Lebesgue space on R+ = (0, ∞) with the norm  · Lp . Let s be a nonnegative integer. Then the Sobolev space W s,p = W s,p (R+ ) is defined by W s,p = {u ∈ Lp ; ∂xk u ∈ Lp for k  s} with the norm  · W s,p . When p = 2, we write H s = W s,2 . Next we introduce weighted spaces. Let w = w(x) > 0 be a weight function defined on [0, ∞) such that w ∈ C 0 [0, ∞). Then, for 1  p < ∞, we denote by Lp (w) the weighted Lp space on R+ equipped with the norm  ∞ 1/p p  u(x) w(x) dx . uLp (w) :=

(1.6)

0

The corresponding weighted Sobolev space W s,p (w) is defined by W s,p (w) = {u ∈ Lp (w); 1,p ∂xk u ∈ Lp (w) for k  s} with the norm  · W s,p (w) . Also, we denote by W0 (w) the completion of C0∞ (R+ ) with respect to the norm  ∞ 1/p p  ∂x u(x) w(x) dx uW 1,p (w) := ∂x uLp (w) = .

(1.7)

0

0

When p = 2, we write H s (w) = W s,2 (w) and H01 (w) = W01,2 (w). In the special case where w = (1 + x)α with α ∈ R, these weighted spaces are abbreviated as   Lpα = Lp (1 + x)α ,    1,p 1,p  Wαs,p = W s,p (1 + x)α , Wα,0 = W0 (1 + x)α ,     1 Hα,0 = H01 (1 + x)α . Hαs = H s (1 + x)α , Let k be a nonnegative integer. Then, for an interval I ⊂ [0, ∞) and a Banach space X on R+ , C k (I ; X) denotes the space of k-times continuously differentiable functions on I with values in X. Finally, letters C and c in this paper denote positive generic constants which may vary from line to line.

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2. Hardy type inequality Hardy’s inequality was first introduced by Hardy [1] and its best possible constant was given by Landau [8]. Here we introduce several Hardy type inequalities which will be used in this paper. The following first one is found in [13] but its best possible constant is not given explicitly there. Proposition 2.1. Let ψ ∈ C 1 [0, ∞) and assume either (1) ψ > 0, ψx > 0 and ψ(x) → ∞ for x → ∞; or (2) ψ < 0, ψx > 0 and ψ(x) → 0 for x → ∞. Then we have ∞

∞ v ψx dx  4 2

0

vx2 ψ 2 /ψx dx

(2.1)

0

for v ∈ C0∞ (R+ ) and hence for v ∈ H01 (w) with w = ψ 2 /ψx . Here 4 is the best possible constant, and there is no function v ∈ H01 (w), v = 0, which attains the equality in (2.1). Proof. Let v ∈ C0∞ (R+ ). A simple calculation gives  2  1 1 v ψ x = v 2 ψx + 2vvx ψ = v 2 ψx + (v + 2vx ψ/ψx )2 ψx − 2vx2 ψ 2 /ψx . 2 2

(2.2)

Integrating (2.2) in x, we obtain ∞

∞ ∞ 2 v ψx dx + (v + 2vx ψ/ψx ) dx = 4 vx2 ψ 2 /ψx dx, 2

0

0

(2.3)

0

which gives the desired inequality (2.1). It follows from (2.3) that the equality in (2.1) holds if and only if v + 2vx ψ/ψx ≡ 0. This gives v = C1 |ψ|−1/2 for some constant C1 . But, if C1 = 0, this v is not in H01 (w) with w = ψ 2 /ψx . In fact, in the case (1), we have vx = − 12 C1 ψ −3/2 ψx and hence ∞ vx2 w dx 0

1 = C12 4

∞

∞ 1 ψ −1 ψx dx = C12 log ψ(x) x=0 = ∞. 4

0

The case (2) can be treated similarly. Thus we conclude that there is no function v ∈ H01 (w), v = 0, which attains the equality in (2.1).

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Finally, we show the best possibility of the constant 4 in (2.1). The following proof is based on the computation in [8]. First, we consider the case (1). Let us fix a > 0. Let  > 0 be a small parameter and put ⎧ ⎨ 0, 0  x < a,  v (x) = (x − a)ψ(x)−1/2− , a < x < a + 1, ⎩ ψ(x)−1/2− , a + 1 < x.

(2.4)

Then we have ∞ a+1 ∞   2 2 −1−2 v ψx dx = (x − a) ψ ψx dx + ψ −1−2 ψx dx a

0

a+1

=: I1 + I2 . Here we see that I1 = O(1) for  → 0 and ∞  1 1 −2 I2 = − ψ(x) = ψ(a + 1)−2 . 2 2 x=a+1 On the other hand, we have ⎧ ⎨ 0, 0  x < a,  vx (x) = ψ(x)−1/2− − (1/2 + )(x − a)ψ(x)−3/2− ψx (x), ⎩ −(1/2 + )ψ(x)−3/2− ψx (x), a + 1 < x.

a < x < a + 1,

Therefore we get ∞ a+1   2 2  −1/2− 2 vx ψ /ψx dx = ψ − (1/2 + )(x − a)ψ −3/2− ψx ψ 2 /ψx dx a

0

∞ + (1/2 + )

2

ψ −1−2 ψx dx =: J1 + J2 .

a+1 1 Here we find that J1 = O(1) for  → 0 and J2 = (1/2 + )2 2 ψ(a + 1)−2 . Consequently, we obtain

∞ 0

1 ψ(a + 1)−2 (v  )2 ψ 2 /ψx dx O(1) + (1/2 + )2 2  ∞x = 1  2 O(1) + 2 ψ(a + 1)−2 0 (v ) ψx dx

=

O() + (1/2 + )2 ψ(a + 1)−2 1 −→ 4 O() + ψ(a + 1)−2

for  → 0. This shows that 4 in (2.1) is the best possible constant.

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In the case (2), to show the best possibility of the constant in (2.1), we may take a test function v  (x) as ⎧ ⎨ 0, 0  x < a,  v (x) = (x − a)(−ψ(x))−1/2− , a < x < a + 1, ⎩ (−ψ(x))−1/2− , a + 1 < x, 2

but we omit the details. This completes the proof of Proposition 2.1. We have the Lp version of Proposition 2.1.

Proposition 2.2. Let ψ be the same as in Proposition 2.1. Let 1 < p < ∞. Then we have ∞

∞ |v| ψx dx  p p

0

p−1

|vx |p |ψ|p /ψx

p

(2.5)

dx

0

for v ∈ C0∞ (R+ ) and hence for v ∈ W0 (w) with w = |ψ|p /ψx . Here p p is the best possible 1,p constant, and there is no function v ∈ W0 (w), v = 0, which attains the equality in (2.5). 1,p

p−1

Proof. Let 1 < p < ∞ and v ∈ C0∞ (R+ ). A simple calculation gives  p  |v| ψ x = |v|p ψx + p|v|p−2 vvx ψ =

1 p p−1  |v| ψx − p p |vx |p |ψ|p /ψx + R, p

(2.6)

where   1 1 p−1 |v|p ψx + p p |vx |p |ψ|p /ψx + p|v|p−2 vvx ψ. R= 1− p p Integrating (2.6) in x, we obtain ∞

∞ |v| ψx dx + p

0

∞ R dx = p

p

0

p−1

|vx |p |ψ|p /ψx

p

dx.

0

Here we see that −p|v|p−2 vvx ψ  p|v|p−1 |vx ||ψ|  (p−1)/p  (p−1)/p  p|vx ||ψ|/ψx = |v|p−1 ψx   1 1 p−1  1− |v|p ψx + p p |vx |p |ψ|p /ψx , p p

(2.7)

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where we have used the Young inequality AB  (1 − 1/p)Ap/(p−1) + (1/p)B p for A = (p−1)/p (p−1)/p and B = p|vx ||ψ|/ψx . Thus we have R  0. This together with (2.7) |v|p−1 ψx gives the desired inequality (2.5). It follows from (2.7) that the equality in (2.5) holds if and only if R ≡ 0. This is the case p−1 where vvx ψ  0 and |v|p ψx ≡ p p |vx |p |ψ|p /ψx . This is equivalent to pvx ψ ≡ −vψx and −1/p for some constant C1 . A simple computation shows that when hence we have v = C1 |ψ| 1,p p−1 C1 = 0, this v is not in W0 (w) with w = |ψ|p /ψx . Thus we conclude that there is no 1,p function v ∈ W0 (w), v = 0, which attains the equality in (2.5). The best possibility of the constant p p is proved in the same way as in the proof of Proposition 2.1. For example, in the case (1), we take the test function as ⎧ ⎨ 0, 0  x < a,  v (x) = (x − a)ψ(x)−1/p− , a < x < a + 1, ⎩ ψ(x)−1/p− , a + 1 < x,

(2.8)

where a > 0 is fixed and  > 0 is a small parameter. Then we see that ∞ 0

p−1 p 1 −p |vx |p |ψ|p /ψx dx O(1) + (1/p + ) p ψ(a + 1) ∞ = 1  p O(1) + p ψ(a + 1)−p 0 |v | ψx dx

=

O() + (1/p + )p ψ(a + 1)−p 1 −→ p O() + ψ(a + 1)−p p

for  → 0. This shows that p p in (2.5) is the best possible constant. Thus the proof of Proposition 2.2 is complete. 2 The following variant of Proposition 2.1 is useful in our application. Proposition 2.3. Let φ ∈ C 1 [0, ∞), φ < 0, φx > 0, and φ(x) → 0 for x → ∞. Let σ ∈ R with σ = 0, and define the weight functions w and w1 by w = (−φ)−σ +1 /φx ,

w1 = (−φ)−σ −1 φx .

(2.9)

Then we have ∞

4 v w1 dx  2 σ

∞

2

0

vx2 w dx

(2.10)

0

for v ∈ H01 (w). Here 4/σ 2 is the best possible constant, and there is no function v ∈ H01 (w), v = 0, which attains the equality in (2.10).

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Proof. Let σ > 0. In this case we put ψ = (−φ)−σ > 0. Then we have ψx = σ (−φ)−σ −1 φx > 0 and ψ(x) → ∞ as x → ∞. This corresponds to the case (1) of Proposition 2.1. Since ψ 2 /ψx = (1/σ )(−φ)−σ +1 /φx , by applying Proposition 2.1, we have ∞ 2

σ

−σ −1

v (−φ)

4 φx dx  σ

0

∞

vx2 (−φ)−σ +1 /φx dx.

0

This gives (2.10) and hence the proof of Proposition 2.3 is complete for σ > 0. When σ < 0, we put ψ = −(−φ)−σ < 0. Then, applying the case (2) of Proposition 2.1, we get the desired conclusion also for σ < 0. This completes the proof of Proposition 2.3. 2 As a simple corollary of Proposition 2.3, we have: Corollary 2.4. Let α ∈ R with α = 1. Then we have vL2



α−2

2 vx L2α |α − 1|

(2.11)

1 = H 1 ((1 + x)α ). Here the constant 2/|α − 1| is the best possible, and there is no for v ∈ Hα,0 0 1 , v = 0, which attains the equality in (2.11). function v ∈ Hα,0

Proof. Let φ = −(1 + x)−1/q with q > 0. Then we see that φ < 0, φx = (1/q)(1 + x)−1/q−1 = (1/q)(−φ)q+1 > 0, and φ(x) → 0 as x → ∞. Now we apply Proposition 2.3. Since w = (−φ)−σ +1 /φx = q(−φ)−σ −q = q(1 + x)σ/q+1 , 1 1 (−φ)−σ +q = (1 + x)σ/q−1 , q q

w1 = (−φ)−σ −1 φx =

(2.12)

we have from (2.10) that 1 q

∞ v (1 + x) 2

σ/q−1

4q dx  2 σ

0

∞ vx2 (1 + x)σ/q+1 dx 0

1 for v ∈ Hσ/q+1,0 . Thus we have

v2L2

σ/q−1



4q 2 vx 2L2 , σ2 σ/q+1

(2.13)

1 for v ∈ Hσ/q+1,0 . This together with σ = (α − 1)q gives the desired inequality (2.11). This completes the proof. 2

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3. Dissipativity of the linearized operator We discuss the dissipativity of the operator L defined by (1.5) in the weighted space L2 (w). For this purpose, we first review the basic properties of the degenerate stationary wave φ(x) (see [9] for the details). Lemma 3.1. Suppose that f (u) satisfies (1.2). Then the stationary wave φ(x), which is a solution of (1.3), verifies the following properties: φ ∈ C ∞ [0, ∞), and −1  φ(x) < 0,

φ(x) → 0 for x → ∞.

φx (x) > 0,

(3.1)

Moreover, we have c(1 + x)−1/q  −φ(x)  C(1 + x)−1/q .

(3.2)

Now, let w > 0 be a weight function depending only on x such that w ∈ C 2 [0, ∞) and we calculate the inner product Lv, v L2 (w) for v ∈ C0∞ (R+ ), where ∞ u, v L2 (w) :=

uvw dx.

(3.3)

0

We multiply (1.5) by v. Then a simple computation gives   1 1 (Lv)v = vvx − f  (φ)v 2 − vx2 − f  (φ)φx v 2 . 2 2 x Multiplying by w, we obtain    1 2 1  2 (Lv)vw = vvx − f (φ)v w − v wx 2 2 x  1  − vx2 w + v 2 wxx + wx f  (φ) − wf  (φ)φx . 2

(3.4)

Now we choose the weight function w and the corresponding w1 in terms of the degenerate stationary wave φ by (2.9), where σ ∈ R. Then we have w = (−φ)−σ +1 /f (φ) by φx = f (φ). Differentiating this expression with respect to x and using φx = f (φ) several times, we find by direct computations that wx = (σ − 1)(−φ)−σ − (−φ)−σ +1 f  (φ)/f (φ), wxx = σ (σ − 1)(−φ)−σ −1 f (φ) − (σ − 1)(−φ)−σ f  (φ)   − (−φ)−σ +1 f  (φ) − f  (φ)2 /f (φ) . Consequently, we arrive at the expression

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wxx + wx f  (φ) − wf  (φ)φx = σ (σ − 1)(−φ)−σ −1 f (φ) − 2(−φ)−σ +1 f  (φ)   = σ (σ − 1) − 2(−φ)2 f  (φ)/f (φ) (−φ)−σ −1 f (φ)   = 2 c1 (σ ) − r(φ) w1 , (3.5) where w1 is given in (2.9) and c1 (σ ) := σ (σ − 1)/2 − q(q + 1), r(u) := (−u)2 f  (u)/f (u) − q(q + 1).

(3.6)

Substituting (3.5) into (3.4) and integrating with respect to x, we get the following conclusion. Claim 3.2. Let φ be the degenerate stationary wave and define the weight functions w and w1 by (2.9) with σ ∈ R. Then the operator L defined in (1.5) verifies ∞ Lv, v L2 (w) = −vx 2L2 (w)

+ c1 (σ )v2L2 (w ) 1

−

v 2 r(φ)w1 dx

(3.7)

0

for v ∈ C0∞ (R+ ) and hence for v ∈ H01 (w), where c1 (σ ) and r(φ) are given in (3.6). To discuss the dissipativity of L, we need to estimate the term r(φ) in (3.7). By straightforward computations, using (3.6) and (1.2), we see that    r(u) = (−u) (−u)g  (u) − 2(q + 1)g  (u) 1 + g(u) . (3.8) This shows that r(u) = O(|u|) for u → 0. In particular, we have r(u) ≡ 0 if g(u) ≡ 0. With these preparations, we have the following result on the characterization of the dissipativity of L. Theorem 3.3. Assume (1.2). Let φ be the degenerate stationary wave and L be the operator defined in (1.5). Let w and w1 be the weight functions in (2.9) with the parameter σ ∈ R. Then we have: (1) Let −2q < σ < 2(q + 1). Then, under the additional assumption that r(u)  0 for −1  u  0, the operator L is uniformly dissipative in L2 (w). Namely, there is a positive constant δ such that   Lv, v L2 (w)  −δ vx 2L2 (w) + v2L2 (w ) (3.9) 1

for v ∈ H01 (w). (2) Let σ > 2(q + 1) or σ < −2q. Then the operator L cannot be dissipative in L2 (w). Namely, we have Lv, v L2 (w) > 0 for some v ∈ H01 (w) with v = 0. Remark 3.4. In (1) of this theorem, we have assumed that r(u)  0 for −1  u  0. In view of (3.8), this additional condition is satisfied if g(u) in (1.2) is of the form g(u) = G(−u), where G (η)  0 and G (η)  0 for 0  η  1. The simplest example of such a g(u) is g(u) = (−u)m with a nonnegative integer m.

S. Kawashima, K. Kurata / Journal of Functional Analysis 257 (2009) 1–19

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Proof. The proof is based on the equality (3.7) in Claim 3.2 and the Hardy type inequality (2.10) in Proposition 2.3. Let −2q < σ < 2(q + 1). This is equivalent to c1 (σ ) < σ 2 /4. Therefore we can choose δ > 0 so small that δ(1 + σ 2 /4)  σ 2 /4 − c1 (σ ). Since r(φ)  0 by the additional assumption on r(u) and since (σ 2 /4)v2L2 (w )  vx 2L2 (w) by the Hardy type inequality (2.10), we have from (3.7) 1 that Lv, v L2 (w)  −vx 2L2 (w) + c1 (σ )v2L2 (w

1)

= −δvx 2L2 (w)

− (1 − δ)vx 2L2 (w)

+ c1 (σ )v2L2 (w ) 1   2 2 2  −δvx L2 (w) − (1 − δ)σ /4 − c1 (σ ) vL2 (w ) 1   2 2  −δ vx L2 (w) + vL2 (w ) 1

for v ∈ C0∞ (R+ ) and hence for v ∈ H01 (w), where we used the fact that (1 − δ)σ 2 /4 − c1 (σ )  δ. This completes the proof of the uniform dissipative case (1). Next we consider the case where σ > 2(q + 1); the case σ < −2q can be treated similarly and we omit the argument in this latter case. When σ > 2(q + 1), we have c1 (σ ) > σ 2 /4. Then we choose δ > 0 so small that c1 (σ )  σ 2 /4 + 3δ. Since r(u) = O(|u|) for u → 0 and φ(x) → 0 for x → ∞, we take a = a(δ) > 0 so large that |r(φ)|  δ for x  a. For this choice of a and for  > 0, we take a test function v  as in (2.4): ⎧ ⎨ 0, 0  x < a,  v (x) = (x − a)(−φ(x))σ (1/2+) , a < x < a + 1, (3.10) ⎩ (−φ(x))σ (1/2+) , a + 1 < x. Then we have  ∞  ∞      2   2    2 v r(φ)w1 dx   δ v w1 dx = δ v  L2 (w ) ,  1   a

0

so that we have from (3.7) that 

Lv  , v 

 L2 (w)

 2   2  −vx L2 (w) + c1 (σ ) − δ v  L2 (w ) . 1

Here, a direct computation shows that   2 v  2

L (w1

a+1  ∞ 2 2σ −1 = (x − a) (−φ) φx dx + (−φ)2σ −1 φx dx ) a

a+1

2σ  1  = O(1) + −φ(a + 1) 2σ  for  → 0, where the term denoted by O(1) depends on δ. Similarly, we have   2 v  2

x L (w)

= O(1) + σ 2 (1/2 + )2

2σ  1  −φ(a + 1) 2σ 

(3.11)

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for  → 0. Consequently, we obtain vx 2L2 (w) v  2L2 (w

=

O(1) + σ 2 (1/2 + )2 2σ1  (−φ(a + 1))2σ  O(1) +

1)

=

1 2σ  (−φ(a

+ 1))2σ 

O() + σ 2 (1/2 + )2 (−φ(a + 1))2σ  σ2 −→ 2σ  4 O() + (−φ(a + 1))

for  → 0. Thus we have vx 2L2 (w) /v  2L2 (w )  σ 2 /4 + δ for a suitably small  = (δ) > 0. 1 Consequently, we have from (3.11) that Lv  , v  L2 (w) v  2L2 (w

1)

−

vx 2L2 (w)

+ c1 (σ ) − δ v  2L2 (w ) 1    − σ 2 /4 + δ + c1 (σ ) − δ  δ.

This completes the proof of the non-dissipative case (2). Thus the proof of Theorem 3.3 is complete. 2 Finally in this section, we consider the special case where g(u) ≡ 0 so that f (u) = q1 (−u)q+1 . In this case the degenerate stationary wave is given explicitly by φ(x) = −(1 + x)−1/q and the operator L in (1.5) is reduced to L0 below: L0 v = vxx +

  v q +1 . q 1+x x

(3.12)

For this simplest case, we have the complete characterization of the dissipativity of the operator L0 . Theorem 3.5. Let αc (q) := 3 + 2/q. Then we have the complete characterization of the dissipativity of the operator L0 given in (3.12): (1) Let −1 < α < αc (q). Then L0 is uniformly dissipative in L2α . Namely, there is a positive constant δ such that   L0 v, v L2α  −δ vx 2L2 + v2L2 α

(3.13)

α−2

1 . for v ∈ Hα,0 (2) Let α = αc (q) or α = −1. Then L0 is strictly dissipative in L2α . Namely, we have 1 with v = 0. L0 v, v L2α < 0 for v ∈ Hα,0 (3) Let α > αc (q) or α < −1. Then L0 cannot be dissipative in L2α . Namely, we have 1 with v = 0. L0 v, v L2α > 0 for some v ∈ Hα,0

S. Kawashima, K. Kurata / Journal of Functional Analysis 257 (2009) 1–19

13

Proof. Consider the case where f (u) = q1 (−u)q+1 with g(u) ≡ 0. In this case we have φ(x) = −(1 + x)−1/q and L = L0 . Moreover, noting that r(u) ≡ 0, we have as a counterpart of (3.7), L0 v, v L2 (w) = −vx 2L2 (w) + c1 (σ )v2L2 (w ) , 1

(3.14)

where w and w1 are the weight functions defined in (2.9) with σ ∈ R, and c1 (σ ) is given in (3.6). In our special case, these weight functions are given explicitly by (2.12), so that we have L0 v, v L2 (w) = q L0 v, v L2

σ/q+1

vx L2 (w) = qvx 2L2

σ/q+1

,

,

vL2 (w1 ) =

1 v2L2 . q σ/q−1

(3.15)

Now we put σ = (α − 1)q. First, let −1 < α < αc (q). This corresponds to the case where −2q < σ < 2(q + 1), for which we have c1 (σ ) < σ 2 /4. Therefore, applying to (3.14) the same arguments as in (1) of Theorem 3.3, we obtain   L0 v, v L2 (w)  −δ vx 2L2 (w) + v2L2 (w ) 1

for some δ > 0. This inequality together with the relations in (3.15) (with σ = (α − 1)q) shows the uniform dissipativity of L0 in L2α . Second, let α = αc (q) or α = −1. This corresponds to the case where σ = 2(q + 1) or σ = −2q. In this case we have c1 (σ ) = σ 2 /4. On the other hand, we have (σ 2 /4)v2L2 (w )  vx 2L2 (w) by the Hardy type inequality (2.10). Consequently, we get from (3.14) that

1

L0 v, v L2 (w)  0. Here the equality holds if and only if (σ 2 /4)v2L2 (w

1)

= vx 2L2 (w) . However, we know

from Proposition 2.3 that such a v = 0 does not exist in H01 (w). Thus we conclude that L0 v, v L2 (w) < 0 for v ∈ H01 (w) with v = 0, which together with (3.15) (with σ = (α − 1)q) proves the strict dissipativity of L0 in L2α . Finally, let α > αc (q) or α < −1. Then we have σ > 2(q + 1) or σ < −2q and hence c1 (σ ) > σ 2 /4. Therefore, applying to (3.14) the same arguments as in (2) of Theorem 3.3, we find that L0 v, v L2 (w) > 0 for some v ∈ H01 (w) with v = 0. This together with (3.15) (with σ = (α − 1)q) gives the desired conclusion of (3) of Theorem 3.5. This completes the proof. 2 4. Nonlinear stability The aim of this section is to prove the following stability result for the nonlinear problem (1.4) that is a refinement of the result in [15]. Theorem 4.1. Assume (1.2). Suppose that v0 ∈ L2α ∩ L∞ for some α with 1  α < αc (q) := 3 + q/2. Then there is a positive constant δ1 such that if v0 L2  δ1 , then the problem (1.4) has 1

a unique global solution v ∈ C 0 ([0, ∞); L2α ∩ Lp ) for each p with 2  p < ∞. Moreover, the solution verifies the decay estimate     v(t) p  C v0  2 + v0 L∞ (1 + t)−α/4−ν (4.1) Lα L for t  0, where 2  p < ∞, ν = (1/2)(1/2 − 1/p), and C is a positive constant.

14

S. Kawashima, K. Kurata / Journal of Functional Analysis 257 (2009) 1–19

A key to the proof of Theorem 4.1 is to show the following space–time weighted energy inequality. Proposition 4.2. Assume the same conditions as in Theorem 4.1. Let v be a solution to the problem (1.4) with the initial data v0 ∈ L2α ∩ L∞ , where 1  α < αc (q) := 3 + 2/q. Then there is a positive constant δ2 such that if v0 L2  δ2 , then we have 1

  v(t)

L21

 Cv0 L2 .

(4.2)

1

Moreover, we have the following space–time weighted energy inequality: 2 v(t)L2 +



γ

(1 + t)

t

β

2  2   (1 + τ )γ vx (τ )L2 + v(τ )L2 dτ β

β−2

0

t  Cv0 2L2 β

+γC

2  (1 + τ )γ −1 v(τ )L2 dτ

(4.3)

β

0

for any γ  0 and β with 0  β  α, where the constant C is independent of γ and β. Proof. The main part of the proof of this proposition is to derive the following space–time weighted energy inequality: 2 v(t)L2 +



γ

(1 + t)

t

β

2  2   (1 + τ )γ vx (τ )L2 + v(τ )L2 dτ β

β−2

0

t  Cv0 2L2 β

+γC

2  γ (1 + τ )γ −1 v(τ )L2 dτ + CSβ (t)

(4.4)

β

0

for any γ  0 and β with 0  β  α, where 1  α < αc (q) := 3 + 2/q, C is a constant independent of γ and β, and

γ Sβ (t) =

t

3  (1 + τ )γ v(τ )L3

(4.5)

dτ.

β−1

0

Once (4.4) is obtained, we can show the desired estimates (4.2) and (4.3) as follows. We observe that γ Sβ (t)  C

t

2    2   (1 + τ )γ v(τ )L2 vx (τ )L2 + v(τ )L2 dτ, 1

0

β

β−2

(4.6)

S. Kawashima, K. Kurata / Journal of Functional Analysis 257 (2009) 1–19

15

which is an easy consequence of the following inequality (see [15] for the details): v3L3

β−1

 CvL2 vL2 1

β−2

  vx L2 + vL2 , β

β−2

where β ∈ R. Now we put γ = 0 and β = 1 in (4.4) and define V (t)  0 by 2  V (t) = sup v(τ )L2 +

t

2

0τ t

1

     vx (τ )2 2 + v(τ )2 2 dτ. L L 1

−1

0

Since S10 (t)  CV (t)3 by (4.6), we get the inequality V (t)2  Cv0 2 2 + CV (t)3 . This can be L1

solved as V (t)  Cv0 L2 , provided that v0 L2 is suitably small. Thus we obtain 1

1

  v(t)2 2 +

t

L1

     vx (τ )2 2 + v(τ )2 2 dτ  Cv0 2 2 , L L L −1

1

(4.7)

1

0

which gives the uniform estimate (4.2). Consequently, we have t Sγβ (t)  Cv0 L2 1

2  2   (1 + τ )γ vx (τ )L2 + v(τ )L2 dτ. β

β−2

0

Substituting this estimate into (4.4) and assuming that v0 L2 is suitably small (say, v0 L2  δ2 ), 1 1 we arrive at the desired energy inequality (4.3). It remains to prove the inequality (4.4). To this end, we recall the following uniform estimate:   v(t)

L∞

 M∞ ,

(4.8)

where M∞ = v0 L∞ + 2. This is an easy consequence of the maximum principle (see [5] for the details). Proof of (4.4) for β = 0. The proof is based on the time weighted L2 energy method. We multiply Eq. (1.4) by v. This yields 

1 2 v 2

 + (F − vvx )x + vx2 + G = 0,

(4.9)

t

where   F = f (φ + v) − f (φ) v −

v

  f (φ + η) − f (φ) dη,

0

v G= 0

   f (φ + η) − f  (φ) dη · φx .

(4.10)

16

S. Kawashima, K. Kurata / Journal of Functional Analysis 257 (2009) 1–19

We note that   1 F = f  (φ)v 2 + O |v|3 , 2

  1 G = f  (φ)φx v 2 + φx O |v|3 2

(4.11)

for v → 0. Also, we observe that    1 (−φ)q+1 1 + O |φ| , q    f  (φ) = (q + 1)(−φ)q−1 1 + O |φ|

φx = f (φ) =

for |φ| → 0 and that f  (φ) > 0 by (1.2). Therefore, noting (3.2) and (4.8), we have from (4.11) that G  c(1 + x)−2 v 2 − C(1 + x)−1−1/q |v|3

(4.12)

for any x ∈ R+ . We integrate (4.9) over R+ and substitute (4.12) into the resulting equality. This gives 1 d v2L2 + vx 2L2 + cv2L2  Cv3L3 . 2 dt −2 −1 We multiply this inequality by (1 + t)γ and integrate with respect t. This yields the desired inequality (4.4) for β = 0. Proof of (4.4) for β > 0. We apply the space–time weighted energy method employed in [15] (see also [3]). Let w > 0 be a smooth weight function depending only on x, which will be specified later. We multiply (4.9) by w, obtaining 

1 2 v w 2

 t

    1 2 1 2 2 v wxx + F wx − Gw = 0. + (F − μvvx )w + v wx + vx w − 2 2 x

(4.13)

Here, using (4.11), we have  1 2 1  v wxx + F wx − Gw = v 2 wxx + wx f  (φ) − wf  (φ)φx + R, 2 2

(4.14)

where R = wx O(|v|3 ) − wφx O(|v|3 ) for v → 0. Notice that the coefficient wxx + wx f  (φ) − wf  (φ)φx in (4.14) is just the same as that appeared in (3.4). Now we choose the weight function w and the corresponding w1 by (2.9) with σ = (β − 1)q, where 0  β  α and 1  α < αc (q) := 3 + 2/q. Then we have (3.5) with σ = (β − 1)q. Substituting these expressions into (4.13) and integrating over R+ , we obtain 1 d v2L2 (w) + vx 2L2 (w) − c1 (σ )v2L2 (w ) + 1 2 dt

∞

∞ v r(φ)w1 dx = 2

0

R dx, 0

(4.15)

S. Kawashima, K. Kurata / Journal of Functional Analysis 257 (2009) 1–19

17

where c1 (σ ) and r(φ) are given in (3.6) with σ = (β − 1)q. Here our weight functions in (2.9) are   w = q(−φ)−σ −q / 1 + g(φ) ,

w1 =

  1 (−φ)−σ +q 1 + g(φ) . q

Therefore, noting (3.2), we see that w ∼ (1 + x)σ/q+1 = (1 + x)β ,

w1 ∼ (1 + x)σ/q−1 = (1 + x)β−2 ,

(4.16)

where the symbol ∼ means the equivalence. This implies that the norms  · L2 (w) and  · L2 (w1 ) are equivalent to  · L2 and  · L2 , respectively. β β−2 We estimate (4.15) similarly as in (1) of Theorem 3.3. To this end, we note that σ1  σ  σ2 , where σ1 = −q and σ2 = (α − 1)q. Since c1 (σ ) < σ 2 /4 for −2q < σ < 2(q + 1) and since −2q < σ1 < σ2 < 2(q + 1), we can choose δ > 0 so small that δ

min

σ1 σ σ2

σ 2 /4 − c1 (σ ) . 2 + σ 2 /4

Notice that this δ is independent of β. For this choice of δ, we take a = a(δ) > 0 so large that |r(φ)|  δ for x  a. Then we have  ∞      2  v r(φ)w1 dx   δv2L2 (w1 ) + Cv2L2 ,   −2 0

where C is a constant satisfying C  (1 + x)2 |r(φ)|w1 for 0  x  a. Also, using the Hardy type inequality (σ 2 /4)v2L2 (w )  vx 2L2 (w) in (2.10), we have 1

vx 2L2 (w) − c1 (σ )v2L2 (w ) = δvx 2L2 (w) + (1 − δ)vx 2L2 (w) − c1 (σ )v2L2 (w ) 1 1   2 2 2  δvx L2 (w) + (1 − δ)σ /4 − c1 (σ ) vL2 (w ) 1

 δvx 2L2 (w)

+ 2δv2L2 (w ) , 1

where we used the fact that (1 − δ)σ 2 /4 − c1 (σ )  2δ. On the other hand, using (4.8), we see that |R|  C(|wx | + wφx )|v|3 . Moreover, a straightforward computation shows that |wx | + wφx  C(1 + x)β−1 . Substituting all these estimates into (4.15), we obtain   1 d v2L2 (w) + δ vx 2L2 (w) + v2L2 (w )  Cv2L2 + Cv3L3 , 1 2 dt −2 β−1

(4.17)

where δ and C are independent of β. We multiply this inequality by (1 + t)γ and integrate with respect to t. By virtue of (4.16), we have

18

S. Kawashima, K. Kurata / Journal of Functional Analysis 257 (2009) 1–19

t

2 v(t)L2 +



γ

(1 + t)

β

2  2   (1 + τ )γ vx (τ )L2 + v(τ )L2 dτ β

β−2

0

t  Cv0 2L2 β

+γC

2  (1 + τ )γ −1 v(τ )L2 dτ β

0

t +C

2  γ (1 + τ )γ v(τ )L2 dτ + CSβ (t),

(4.18)

−2

0

where the constant C is independent of γ and β. Here the third term on the right-hand side of (4.18) was already estimated by (4.4) with β = 0. Hence we have proved (4.4) also for 0 < β  α. This completes the proof of Proposition 4.2. 2 Proof of Theorem 4.1. The proof is essentially the same as that of Theorem 3.3 of [15] so that we only give an outline of the proof of the decay estimate (4.1). First, we show the following L2 decay estimate by applying the induction argument to the space–time weighted energy inequality (4.3). 2 (1 + t) v(t)L2 

j

α−2j

t +

2  (1 + τ )j vx (τ )L2

α−2j

2  + v(τ )L2

α−2j −2



dτ  Cv0 2L2

α

(4.19)

0

for each integer j with 0  j  [α/2]. To see this, we put γ = j and β = α − 2j in (4.3), obtaining t

2 (1 + t) v(t)L2 

j

α−2j

+

2  (1 + τ )j vx (τ )L2

α−2j

2  + v(τ )L2

α−2j −2



dτ

0

t  Cv0 2L2 α−2j

+ jC

2  (1 + τ )j −1 v(τ )L2

(4.20)

dτ.

α−2j

0

Then, applying the induction with respect to the integer j with 0  j  [α/2], we obtain (4.19). On the other hand, when α/2 is not an integer, we have 2  (1 + t)γ v(t)L2 +

t

2  2   (1 + τ )γ vx (τ )L2 + v(τ )L2 dτ −2

0

 Cv0 2L2 (1 + t)γ −α/2 . α

(4.21)

for any γ with γ > α/2. This can be shown by using (4.3) with γ > α/2, β = 0 and (4.19) with j = [α/2] together with Nishikawa’s technique in [12]. For the details, see the proof of Proposition 2.6 of [15].

S. Kawashima, K. Kurata / Journal of Functional Analysis 257 (2009) 1–19

Thus we have shown the following L2 decay estimate:   v(t) 2  Cv0  2 (1 + t)−α/4 . Lα L

19

(4.22)

The desired Lp decay estimate (4.1) is then obtained by the time weighted Lp energy method. In fact, under the additional smallness condition on vL2 , we have 1

p v(t)Lp +



γ

(1 + t)

t

p  2    (1 + τ )γ  |v|p/2 x (τ )L2 + v(τ )Lp dτ −2

0

 Cv0 Lp + Cv0 L2 (1 + t)γ −(α/4+ν)p , p

p

(4.23)

α

where 2  p < ∞, ν = (1/2)(1/2 − 1/p) and γ > (α/4 + ν)p. We omit the details and refer the reader to the proof of Theorem 2.3 of [15]. This completes the proof of Theorem 4.1. 2 References [1] G.H. Hardy, Note on a theorem of Hilbert, Math. Z. 6 (1920) 314–317. [2] Y. Kagei, S. Kawashima, Stability of planar stationary solutions to the compressible Navier–Stokes equation on the half space, Comm. Math. Phys. 266 (2006) 401–430. [3] S. Kawashima, A. Matsumura, Asymptotic stability of traveling waves solutions of systems for one-dimensional gas motion, Comm. Math. Phys. 101 (1985) 97–127. [4] S. Kawashima, S. Nishibata, M. Nishikawa, Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane, Discrete Contin. Dyn. Syst. Suppl. (2003) 469–476. [5] S. Kawashima, S. Nishibata, M. Nishikawa, Lp energy method for multi-dimensional viscous conservation laws and application to the stability of planar waves, J. Hyperbolic Differ. Equ. 1 (2004) 581–603. [6] S. Kawashima, S. Nishibata, M. Nishikawa, Asymptotic stability of stationary waves for multi-dimensional viscous conservation laws in half space, preprint, 2004. [7] S. Kawashima, S. Nishibata, P. Zhu, Asymptotic stability of the stationary solution to the compressible NavierStokes equations in the half space, Comm. Math. Phys. 240 (2003) 483–500. [8] E. Landau, A note on a theorem concerning series of positive terms, J. London Math. Soc. 1 (1926) 38–39. [9] T.-P. Liu, A. Matsumura, K. Nishihara, Behavior of solutions for the Burgers equations with boundary corresponding to rarefaction waves, SIAM J. Math. Anal. 29 (1998) 293–308. [10] A. Matsumura, K. Nishihara, Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity, Comm. Math. Phys. 165 (1994) 83–96. [11] T. Nakamura, S. Nishibata, T. Yuge, Convergence rate of solutions toward stationary solutions to the compressible Navier–Stokes equation in a half line, J. Differential Equations 241 (1) (2007) 94–111. [12] M. Nishikawa, Convergence rate to the traveling wave for viscous conservation laws, Funkcial. Ekvac. 41 (1998) 107–132. [13] B. Opic, A. Kufner, Hardy-Type Inequalities, Pitman Res. Notes in Math. Ser., vol. 219, Longman Scientific & Technical, 1990. [14] Y. Ueda, Asymptotic stability of stationary waves for damped wave equations with a nonlinear convection term, Adv. Math. Sci. Appl. 18 (1) (2008) 329–343. [15] Y. Ueda, T. Nakamura, S. Kawashima, Stability of degenerate stationary waves for viscous gases, Arch. Ration. Mech. Anal., in press. [16] Y. Ueda, T. Nakamura, S. Kawashima, Stability of planar stationary waves for damped wave equations with nonlinear convection in multi-dimensional half space, Kinetic Related Models 1 (2008) 49–64.

Journal of Functional Analysis 257 (2009) 20–46 www.elsevier.com/locate/jfa

The planar algebra of group-type subfactors ✩ Dietmar Bisch, Paramita Das, Shamindra Kumar Ghosh ∗ Vanderbilt University, Department of Mathematics, SC 1326, Nashville, TN 37240, USA Received 11 August 2008; accepted 25 March 2009

Communicated by D. Voiculescu

Abstract If G is a countable, discrete group generated by two finite subgroups H and K and P is a II1 factor with an outer G-action, one can construct the group-type subfactor P H ⊂ P  K introduced by Haagerup and the first author to obtain numerous examples of infinite depth subfactors whose standard invariant has exotic growth properties. We compute the planar algebra of this subfactor and prove that any subfactor with an abstract planar algebra of “group type” arises from such a subfactor. The action of Jones’ planar operad is determined explicitly. © 2009 Elsevier Inc. All rights reserved. Keywords: Planar algebra; Planar operad; Subfactor; IRF model

1. Introduction The technique of composing subfactors pioneered in [4] led to a zoo of exotic subfactors of the hyperfinite II1 factor. In particular, the first examples of irreducible, amenable subfactors that are not strongly amenable (in the sense of [16]) were constructed in this way in [4]. The idea is simple: Let H and K be two finite groups with an outer action on a II1 factor P (e.g. the hyperfinite II1 factor) and consider the composition of the fixed-point subfactor P H ⊂ P with the crossed product subfactor P ⊂ P  K to obtain, what we will call here, a group-type subfactor P H ⊂ P  K. If P is hyperfinite, analytical properties of this subfactor, such as amenability ✩

The authors were supported by NSF under Grant No. DMS-0301173.

* Corresponding author.

E-mail addresses: [email protected] (D. Bisch), [email protected] (P. Das), [email protected] (S.K. Ghosh). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.03.014

D. Bisch et al. / Journal of Functional Analysis 257 (2009) 20–46

21

and property (T) in the sense of Popa [16,18], were proved to be equivalent to the corresponding properties (amenability, property (T) in the sense of Kazhdan) of the group G generated by H and K in the outer automorphism group of P [4,7]. Note that a group-type subfactor is obtained from the two fixed-point subfactors P H ⊂ P and P K ⊂ P by performing the basic construction of [10] to one of them. A group-type subfactor is therefore an invariant for the relative position of the two fixed-point subfactors. For more on this, see [12]. The main invariant for a subfactor is the so-called standard invariant (see for instance [9,8,13]). It is a very sophisticated mathematical object that can be portrayed in a number of seemingly quite different ways. For example, it has descriptions as a certain category of bimodules ([14], see also [3]), as a lattice of algebras [17], or as a planar algebra [11]. Jones’ planar algebra technology has become a very efficient tool to capture and analyze the standard invariant of a subfactor. Composition of subfactors was the motivating idea that led to the results in [5]. It was proved there that two standard invariants without extra structure, i.e. consisting of just the Temperley– Lieb algebras, can always be composed freely to form a new standard invariant – namely the one generated by the Fuss–Catalan algebras of [5]. This concept of free composition was then pushed much further in [6], where it is shown that any two planar algebras arising from subfactors can be composed freely to form a new subfactor planar algebra. The principal graphs of a subfactor encode the algebraic information contained in the standard invariant [9], and their structure determines the growth properties of the invariant. It was shown in [4] that subfactors whose standard invariants have very exotic growth properties exist by constructing concrete group-type subfactors. The principal graphs of these subfactors were computed there. In this paper we go a step further and give a concrete description of the standard invariant (or equivalently the planar algebra) of the group-type subfactors. We concentrate on the case when the group G, generated by H and K in the automorphism group of P , has an outer action on P . Note that any group G generated by two finite subgroups H and K has such an action on the hyperfinite II1 factor (for instance by a Bernoulli shift action). We find a description of the planar algebra of these group-type subfactors that is reminiscent of an IRF (interaction round a face) model in statistical mechanics. The general case, where G is generated by H and K in the outer automorphism group of P , and hence may or may not lift to an action of G on P , is much more elaborate and will be treated in a separate paper. Here is a more detailed outline of the sections of this paper. We review in Section 2 the basic notions of planar algebras. In Section 3, we define an abstract planar algebra P associated to a countable, discrete group G and two of its finite subgroups H and K which generate G. The vector spaces underlying the planar algebra are spanned by alternating words in H and K that multiply to the identity element. The action of Jones’ planar operad is given explicitly by a particular labelling of planar tangles that can be viewed as an IRF-like model. It takes some work to show that the action of planar tangles is well defined and preserves composition. The latter is achieved by showing that the action respects composition with certain elementary tangles that generate any annular tangle. We then analyze the filtered ∗-algebra structure of P and determine the action of Jones projection and conditional expectation tangles. In Section 4 we compute the basic construction and higher relative commutants of a grouptype subfactor P H ⊂ P  K. We exhibit a nice basis which is used in Section 5 to prove that the abstract group-type planar algebra of Section 3 is indeed isomorphic to the concrete one computed in Section 4. Moreover, we prove in Section 5 that if the standard invariant of an

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D. Bisch et al. / Journal of Functional Analysis 257 (2009) 20–46

arbitrary subfactor is isomorphic to a group-type planar algebra, then the subfactor is indeed of group-type. This is proved using results on intermediate subfactors from [2,6,1]. 2. Planar algebra basics In this section, we will give a very brief overview of planar algebras; the reader is encouraged to see [11] for details. Let us first describe the key ingredients that constitute a planar tangle. • There is an external disc, several (possibly none) internal discs and a collection of disjoint smooth curves. • To each disc – internal or external, we attach a non-negative integer. This integer will be referred to as the ‘color’ of the disc. If a disc has color k > 0, there will be 2k points on the boundary of the disc marked 1, 2, . . . , 2k counted clockwise, starting with a distinguished marked point, which is decorated with ‘∗’. A disc having color 0 will have no marked points on its boundary. • Each of the curves is either closed, or joins a marked point on the boundary of a disc to another such point, meeting the boundary of the disc transversally. Each marked point must be the endpoint of exactly one curve. • The whole picture has to be planar, in the sense that there should be no crossing of curves or overlapping of discs. • Finally, we will not distinguish between pictures obtained from one another by planar isotopy preserving the ∗’s. The data of such a picture will be termed as a planar k-tangle, where k refers to the color of its external disc. Remark 2.1. We can induce a black-and-white shading on the complement of the union of the internal discs and curves in the external disc by specifying that the region between the last and first marked points be left unshaded. This leaves a scope for ambiguity in the case of 0-discs, thus, 0-discs may be of two types depending on whether the region surrounding their boundary is shaded or unshaded. Given a planar k-tangle T – one of whose internal discs have color ki – and a ki -tangle S, one can define the k-tangle T ◦i S by isotoping S so that its boundary, together with the marked points and the ‘∗’ coincides with that of Di and then erasing the boundary of Di . The collection of tangles – along with the composition defined thus – is called the colored operad of planar tangles. A planar algebra is a collection of vector spaces {Pk }k0 such that every k-tangle T that has b internal discs D1 , D2 , . . . , Db having colors k1 , k2 , . . . , kb respectively gives rise to a multilinear map ZT : Pn1 × Pn2 × · · · × Pnb → Pn0 . The collection of maps is required to be compatible with substitution of tangles and renumbering of internal discs. 3. An abstract planar algebra In this section we will abstractly define a planar algebra which will be identified in Section 5 to be the one corresponding to the group-type subfactor P H ⊂ P  K of [4].

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23

Fig. 1. Example of faces in a tangle.

Let G be a group generated by two of its finite subgroups H and K and e denote the identity element of G. Let us define ⎧ if n = 0, ⎨ {e} K × H × K × H × · · · Sn =    if n  1, ⎩ S=



(n factors)

Sn ,

n0



Ln =

K, H,

if n is even, otherwise.

Let μ : S → G be the multiplication map. With the above notation, we are ready to define the planar algebra but first we would need some terminology. Terminology. By a face in an unlabelled tangle T , we will mean a connected component of D0 \  [( bi=1 Di ) ∪ S] where D0 is the external disc, Di is the ith internal disc for i = 1, 2, . . . , b and S is the set of strings of (an element in the isotopy class of) T . By an opening in a tangle, we will mean the subset of the boundary of a disc lying between two consecutive marked points. An opening will be called internal (resp., external) if it is a subset of the boundary of the internal (resp., external) disc. Note that the boundary of a generic face may be disconnected due to the presence of loops or networks inside it (see Fig. 1). The set of connected components of the boundary of each face will have a single outer component and several (possibly none) inner component(s). Definition 3.1. A state f on a tangle T is a function f : {all openings in T } → H  K such that following holds:

(i)

f (α) ∈

K, H,

if the face containing α is shaded, otherwise.

(ii) Triviality on the outer component of the boundary of a face: Let α1 , α2 , . . . , αm be the openings on the outer component of the boundary of a face counted clockwise, then we must have

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Fig. 2. An example of a state on an unlabelled tangle.

f (α1 )η1 f (α2 )η2 · · · f (αm )ηm = e where

ηi =

+1, if αi is an external opening, −1, otherwise.

(iii) Triviality on internal discs: f induces a map ∂f : {D0 , D1 , . . . , Db } → n0 Sn defined by

(i) 

(i)  (i)  ∂f (Di ) = f α1 , f α2 , . . . , f α2ni ∈ S2ni (i)

(i)

(i)

(i)

where α1 , α2 , . . . , α2ni are consecutive openings counted clockwise such that α1 is the opening between the first and the second marked points of ∂Di . We demand that μ(∂f (Di )) = e, for all i = 1, 2, . . . , b. Note that triviality on the external disc and every boundary component of every face follows (see Remark 3.2 for a proof of this fact). The above definition also applies to networks (positively or negatively oriented). Fig. 2 illustrates conditions (ii) and (iii) of Definition 3.1; triviality on internal discs gives b1 b2 b3 b4 = c1 c2 c3 c4 = d1 d2 d3 d4 d5 d6 d7 d8 = f1 f2 = e and triviality on the outer component of the boundaries of faces gives a1 b1−1 = a2 b2−1 = a3 d5−1 d1−1 b3−1 = c4−1 f2−1 = c2 = d2−1 d4−1 = d3 = a4 a6 b4−1 d8−1 d6−1 = d7 = a5 = e. Note that the above relations also imply a1 a2 a3 a4 a5 a6 = e, and c1−1 c3−1 f1−1 = e.

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As the computation involving Fig. 2 suggests, triviality on inner boundaries of a face and on the external disc are actually consequences of the definition of a state. This is made precise in the following remark. Remark 3.2. Let f be a state on a tangle or a network. Then the following conditions hold: (ii) Triviality on every inner component of the boundary of a face: For every inner component of the boundary of a face with openings α1 , α2 , . . . , αm counted clockwise, we have f (α1 )f (α2 ) · · · f (αm ) = e. (iii) Triviality on the external disc: This is just a restatement of condition (iii) in Definition 3.1 applied to the external disc only if its color is greater than zero. We prove the above remark using planar graphs. Without loss of generality, we may start with a tangle or a network which is connected. By a network or a tangle (with non-zero color on its external disc) being connected, we mean that the union of the boundaries of all the discs and strings is a connected set; a 0-tangle is said to be connected if the network obtained after removing its external disc is connected. To each tangle or network, we associate a planar graph whose vertex set is the set of all marked points on the internal and external discs, and the edges are the openings and the strings. Further, we make this graph a directed one in such a way that the directions on the edges arising from the openings are induced by clockwise orientation on the boundary of the discs, and the remaining edges (coming from the strings) are free to have any direction. Any state f assigns group elements to edges arising from openings; we label each of the remaining edges by e and that is why we did not put any restriction on the direction of such edges. Note that the definition of the state implies the following condition on the group labelled planar directed graph: Triviality on the boundary each bounded face of the graph1 : If g1 , g2 , . . . , gm are the group elements assigned to consecutive edges around a face read clockwise, then η

η

ηm =e g1 1 g2 2 · · · gm

where

ηi =

+1, −1,

if ith edge induces clockwise orientation in the face, otherwise.

To establish Remark 3.2, it is enough to prove triviality on the boundary of the unbounded face. To see this, we first consider a pair of bounded faces which have at least one vertex or edge in common. If this pair is considered as a separate graph, then using triviality on each face, it is easy to check triviality on the boundary of the unbounded face of this pair. One can then use this fact inductively to deduce the desired result for the whole graph. Next, we give the definition of the planar algebra. Let the set of states on a tangle T be denoted by S(T ). 1 Since we started with a connected tangle or network, the associated planar graph will be connected; in particular, boundary of each face of the graph will be connected.

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Fig. 3. Assigning scalars to maxima or minima.

The vector spaces. For n  0, define Pn = C{s ∈ S2n : μ(s) = e}. Action of tangles. Let T be an unlabelled tangle with (possibly zero) internal disc(s) D1 , D2 , . . . , Db and external disc D0 where the color of Di is ni . Then T defines a multilinear map, denoted by ZT : Pn1 × Pn2 × · · · × Pnb → Pn0 . We will define ZT (s1 , s2 , . . . , sb ) ∈ Pn0 , where si ∈ S2ni such that μ(si ) = e. In fact, we will just prescribe the coefficient of s0 ∈ S2n0 (such that μ(s0 ) = e) in the expansion of ZT (s1 , s2 , . . . , sb ) in terms of the canonical basis mentioned above. We choose and fix a representative in the isotopy class of T and call it the standard form of T , denoted by T˜ . It is assumed to satisfy the following properties: • T˜ is in rectangular form – meaning that – all of its discs are replaced by boxes and it is placed in R2 in such a way that the boundaries of the boxes are parallel to the co-ordinate axes. • The first marked point on the boundary of each box is on the top left corner. • The collection of strings have finitely many local maxima or minima. • The external box can be sliced by horizontal lines in such a way that each maxima, minima, internal box is in a different slice. To each local maximum or minimum of a string with end-points, we assign a scalar according to Fig. 3. Let p(T ) denote the product of all scalars arising from the local maxima or minima and n+ (T ) (resp., n− (T )) be the number of non-empty connected positively (resp., negatively) oriented network(s) in the tangle T˜ . Then, the coefficient ZT (s1 , s2 , . . . , sb )|s0  of s0 in ZT (s1 , s2 , . . . , sb ) is given by   p(T )|H |n+ (T ) |K|n− (T )  f ∈ S(T ): ∂f (Di ) = si for all i = 0, 1, . . . , b . Note that there could be several standard form representatives for T . However, one standard form representative for T can be transformed into another by a finite sequence of moves of the following three types:

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I. Horizontal or vertical sliding of boxes. II. Wiggling of the strings. III. Rotation of an internal box by a multiple of 360◦ . It is easy to check that the above three moves do not alter the number of connected networks and keep |{f ∈ S(T ): ∂f (Di ) = si for all i = 0, 1, . . . , b}| unaltered. So, it remains to show that p(T ) is unchanged under the moves. Type I moves are the easiest because they do not generate any new local maxima or minima. In each of the moves of type II and III, there arises pair(s) of local maximum and minimum in such a way that the two scalars assigned to each pair are inverses of each other; as a result, p(T ) remains unchanged. Action of the tangles preserve composition. For S an n0 -tangle with internal discs D1 , D2 , . . . , Db having colors n1 , n2 , . . . , nb respectively, and T an nj -tangle for some j ∈ {1, 2, . . . , b}, we would like to show ZS◦Dj T = ZS ◦ (idPn1 × · · · ZT · · · × idPnb ). For this, we first identify a set E of annular tangles (with the distinguished internal disc as D1 ) which we call elementary tangles, namely: (i) Capping tangles:

with col(D1 ) = n  1, col(D0 ) = n − 1, and 1  i  2n − 1 (col(D1 ) = 1 just means that there are no strings connecting the internal disc to the external disc). (ii) Cap inclusion tangles:

with col(D1 ) = n  0, col(D0 ) = n + 1, and 1  i  2n + 1.

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If i = 1 in (i) or (ii) then the pictures should be interpreted as simply not having the bunch of (i − 1) straight strings joining the marked points of the internal disc and the corresponding points of the external disc. (iii) Left inclusion tangles:

with col(D1 ) = n  0, col(D0 ) = n + 2. (iv) Disc inclusion tangle:

(iv) Disc inclusion tangle:

with col(D1 ) = col(D0 ) = n  0, col(D2 ) = q, where p  0, q  0, r  0 such that p + q + r = n and p is even. Note that any annular tangle can be expressed as a composition of the elementary tangles. To see this, express the annular tangle in standard form and cut it into horizontal strips each of which contains at most one internal disc or one local maxima or minima. Now, the strip containing the distinguished internal disc inside the annular tangle can be obtained by composition of elementary tangles of type (iii) and type (ii) (more specifically, the inclusion tangles); one can then glue the other strips consecutively one after the other along the lines of cutting to get back the original tangle. Each such gluing operation is given by composition of an elementary tangle of type (i), (ii), (iv) or (iv) . So, to prove that the action of the tangles preserve composition, it is enough to prove ZE◦D1 T = ZE ◦ ZT (resp., ZE◦D1 T = ZE ◦ (ZT × idPq )) for any tangle T and any E ∈ E of type (i), (ii) or (iii) (resp. (iv) or (iv) ) whenever the composition makes sense.

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We fix an n-tangle T with internal discs D1 , D2 , . . . , Db with colors n1 , n2 , . . . , nb respectively, and an n0 -tangle E ∈ E such that both T and E are in standard forms and color of D1 in E is n. Let us consider the standard form on E ◦D1 T induced by the standard forms of E and T . Our goal is to show:  ZE◦D1 T (s1 , s2 , . . . , sb )|s0 =







  ZE (s)|s0 ZT (s1 , s2 , . . . , sb )|s

s∈S2n s.t. μ(s)=e

if E is of type (i), (ii) or (iii), and  ZE◦D1 T (s1 , s2 , . . . , sb , t)|s0 =







  ZE (s, t)|s0 ZT (s1 , s2 , . . . , sb )|s

s∈S2n s.t. μ(s)=e

if E is of type (iv) or (iv) where sj ∈ S2nj for 0  j  b and t ∈ Sq such that μ(sj ) = e = μ(t) for all j . An interesting situation arises when we pick elementary tangles of type (i), since composition of tangles in this case may lead to a change in the number of connected networks. The reasoning in the other cases is either similar or straightforward. For E being type (i) elementary tangle, the above equation is equivalent to:  ⎧ ⎫  ∂f (Dj ) = sj ⎬ ⎨    p(E ◦D1 T )|H |n+ (E◦D1 T ) |K|n− (E◦D1 T )  f ∈ S(E ◦D1 T )  for 1  j  b,   ∂f (D0 ) = s0 ⎭ ⎩ = p(E)p(T )|H |n+ (E)+n+ (T ) |K|n− (E)+n− (T ) ×

 s∈S2n s.t. μ(s)=e

 

   f ∈ S(E)  ∂f (D1 ) = s  ∂f (D0 ) = s0 

 ⎧ ⎫  ∂f (Dj ) = sj ⎬  ⎨      f ∈ S(T )  for 1  j  b,     ⎩  ∂f (D ) = s ⎭  0

where D0 denotes the external disc of T . First, observe that p(E ◦D1 T ) = p(E)p(T ). To show the equality of the remaining scalars, we consider the following two cases. Case 1. The string which connects the ith and the (i + 1)th points on D1 in E, does not produce any new network in E ◦D1 T other than those that are already present in T . Clearly, n (E ◦D1 T ) = n (T ) (since no new network appears in E ◦D1 T ) and n (E) = 0 for  ∈ {+, −}. A typical example of such a case can be viewed in the following picture where we label the openings on the internal discs of T by group elements coming from the coordinates of sj for 1  j  b, and the openings on D0 of E by coordinates of s0 = (g1 , g2 , . . . , g2n−2 ). Using triviality on the boundary of faces in the definition of a state, we get   



         f ∈ S(E)  ∂f (D1 ) = s  = 0 ⇔  f ∈ S(E)  ∂f (D1 ) = s  = 1  ∂f (D0 ) = s0   ∂f (D0 ) = s0    

⇔ s = g1 , . . . , gi−2 , (gi−1 g), e, g −1 , gi , . . . , g2n−2 ∈ S2n for some g ∈ Li .

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Fig. 4.

Define s g = (g1 , . . . , gi−2 , (gi−1 g), e, g −1 , gi , . . . , g2n−2 ) for g ∈ Li . So, it is enough to check   ⎫ ⎧ ⎫ ⎧  ∂f (Dj ) = sj ⎬  ∂f (Dj ) = sj ⎬  ⎨     ⎨    f ∈ S(T )  for 1  j  b,  .  f ∈ S(E ◦D T )  for 1  j  b,  = 1      ⎩  ∂f (D ) = s g ⎭  ∂f (D0 ) = s0 ⎭ g∈Li ⎩  0 Carefully observing Fig. 4 and using triviality on the boundary of faces once again, we get ⎧  ⎫ ⎨  ∂f (Dj ) = sj ⎬     f ∈ S(E ◦D T )  for 1  j  b,  = 0, equivalently, equals to 1 1 ⎩     ∂f (D0 ) = s0 ⎭  ⎧ ⎫  ∂f (Dj ) = sj ⎬ ⎨    ⇒  f ∈ S(T )  for 1  j  b,  = δg,bη1 bη2 ···bηl l 1 2  ∂f (D ) = s g ⎭ ⎩ 0 where ηj = ±1 according as the corresponding opening is external or internal. Conversely, if  ⎧ ⎫  ∂f (Dj ) = sj ⎬  ⎨    f ∈ S(T )  for 1  j  b,   ⎩   ∂f (D ) = s g ⎭ g∈Li  0

η

η

η

is non-zero, then it has to equal to 1 since g must be b1 1 b2 2 · · · bl l by triviality on the boundary of faces in T ; from the unique state on T which makes the above sum non-zero, one can easily induce a well-defined state on E ◦D1 T , and hence  ⎫ ⎧  ∂f (Dj ) = sj ⎬ ⎨     f ∈ S(E ◦D T )  for 1  j  b,  = 1. 1   ⎩  ∂f (D0 ) = s0 ⎭  This completes the proof of Case 1.

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31

Fig. 5.

Case 2. The string which connects the ith and the (i + 1)th points on D1 in E, produces a new (connected) network in E ◦D1 T other than those that are already present in T . First, let us assume that the new network is positively oriented, equivalently, i is odd. Clearly, n− (E ◦D1 T ) = n− (T ) (since no new negatively oriented network appears in E ◦D1 T ), and n+ (E ◦D1 T ) = n+ (T ) + 1. Further, assume that col(D0 )  1. In this case, n (E) = 0 for  ∈ {+, −}. A typical example of this case can be viewed in the following picture where we label the openings on the internal discs of T by group elements coming from the coordinates of sj for 1  j  b, and the openings on D0 of E by coordinates of s0 = (g1 , g2 , . . . , g2n−2 ). So, in this case, it is enough to check  ⎧  ⎫ ⎧ ⎫  ∂f (Dj ) = sj ⎬ ⎨  ∂f (Dj ) = sj ⎬       ⎨   f ∈ S(T )  for 1  j  b,  . |H |  f ∈ S(E ◦D1 T )  for 1  j  b,  =  ⎩  ⎩ ⎭ ⎭  ∂f (D ) = s g   ∂f (D0 ) = s0  g∈H   0

If ⎧ ⎨

 ⎫  ∂f (Dj ) = sj ⎬  f ∈ S(E ◦D1 T )  for 1  j  b, ⎩  ∂f (D0 ) = s0 ⎭ η

η

is non-empty (equivalently, singleton), from Fig. 5, we have a1 a2 · · · ak = e and gi−1 b1 1 b2 2 η · · · bl l = e where ηj = ±1 according as the corresponding opening is external or internal. For any g ∈ H , define f g by setting ∂f (Dj ) = sj for 1  j  b and ∂f (D0 ) = s g . To check whether f g is a state, we consider the face in T appearing in Fig. 5; triviality on the boundary of this face η η η is given by the equation g −1 b1 1 b2 2 · · · bl l (gi−1 g)a1 a2 · · · ak = e which indeed holds. Triviality on all other discs or faces is induced by the existence of the state on E ◦D1 T . Thus,  ⎧ ⎫  ∂f (Dj ) = sj ⎬ ⎨     f ∈ S(T )  for 1  j  b,  = 1  ⎩   ∂f (D ) = s g ⎭  0

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for all g ∈ H . Conversely, if the right-hand side is non-zero, then there exists g ∈ H such that η η η f g (defined earlier) is a state on T . Analyzing Fig. 5, we get g −1 b1 1 b2 2 · · · bl l (gi−1 g)ak−1 · · · a2−1 a1−1 = e. Note that the opening between the ith and the (i + 1)th points of the disc D0 of T , is assigned e. Now, if we consider the network appearing in Fig. 5 separately, then we have triviality on each of its internal faces and discs (induced by f g being a state); by Remark 3.2(ii) , we also have a1 a2 · · · ak = e (triviality on the internal boundary of the external face of the network). This η η η implies b1 1 b2 2 · · · bl l gi−1 = e; as a result, f h is a state for every h ∈ H . So, if right-hand side is η η η non-zero, then it has to be |H |; moreover, we get b1 1 b2 2 · · · bl l gi−1 = e which plays an important role in showing that ⎧  ⎫ ⎨  ∂f (Dj ) = sj ⎬     f ∈ S(E ◦D T )  for 1  j  b,  = 1 ⎩   0 (equivalently, equals to 1). ⎭   ∂f (D0 ) = s0  This finishes the proof for the case where i is odd. For i even, the proof is exactly similar, except that one has to interchange |H | and |K|. The subcase that deserves separate treatment is when col(D0 ) = 0. In this case, n+ (E) = 1 and |{f ∈ S(E): ∂f (D1 ) = s}| = δs,(e,e) . Therefore, it is enough to show 

    f ∈ S(E ◦D T )  ∂f (Dj ) = sj 1   for 1  j  b

 ⎧ ⎫  ∂f (Dj ) = sj  ⎨ ⎬     =  f ∈ S(T )  for 1  j  b, .   ⎩   ∂f (D ) = (e, e) ⎭  0

The proof of the equality of the two sides is similar and is left to the reader. We now analyze the filtered ∗-algebra structure of P and the action of Jones projection tangles and conditional expectation tangles which will be useful in Section 5 to show that P is isomorphic to the planar algebra arising from a group-type subfactor. We start with laying some notations. Define ⎧ if n = 0, ⎨ {e} K × H × K if n  1, Sn = ·· · × H ×  ⎩ (n factors)

⎧ if n = 0, ⎨ {e} H × K × H × K × · · · if n  1. Tn =    ⎩ (n factors)

Define : Sn → Sn by (s1 , s2 , . . . , sn )= (sn−1 , . . . , s2−1 , s1−1 ) for (s1 , s2 , . . . , sn ) ∈ Sn and let : Sn → Sn denote its inverse. Remark 3.3. We describe below the main structural features of the planar algebra P . (i) Identity:

e 1Pn = 

s, e) s∈Sn−1 (s, e,

if n = 0, if n  1.

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(ii) ∗-structure: Define ∗ on P by defining on the basis as

−1 −1  s ∗ = s2n−1 , . . . , s2−1 , s1−1 , s2n where s = (s1 , s2 , . . . , s2n ) ∈ S2n such that μ(s) = e for n  1, and then extend conjugate linearly. Clearly, ∗ is an involution. One also needs to verify whether the action of a tangle T preserves ∗, that is, ZT ∗ ◦ (∗ × · · · × ∗) = ∗ ◦ ZT ; in particular, ZT ∗ (s1∗ , . . . , sb∗ )|s0∗  =

ZT (s1 , . . . , sb )|s0 . It is enough to check this equation for the cases when T has no internal disc or closed loops, and when T is an elementary tangle. The actual verification in each of these cases is completely routine and is left to the reader. (iii) Multiplication: (a1 , l1 , b1 , h1 ) · (a2 , l2 , b2 , h2 ) = δb1 ,a2 (a1 , l1 l2 , b2 , h2 h1 ) where ai ∈ Sn−1 , bi ∈ S n−1 , li ∈ Ln−1 , hi ∈ H such that μ(ai , li , bi , hi ) = e for i = 1, 2 and n  1 (where we consider the elements ai and bi to be void in the case of n = 1). (iv) Inclusion: 

Pn  s →

(s1 , s2 , . . . , sn−1 , l1 , e, l2 , sn+1 , . . . , s2n ) ∈ Pn+1

l1 ,l2 ∈Ln−1 such that l1 l2 =sn

where s = (s1 , s2 , . . . , s2n ) ∈ S2n such that μ(s) = e for n  1. (v) Jones Projection Tangle: For P2 ,

 =

 |K|  e, h, e, h−1 |H | h∈H

and for Pn+1 for n > 1,

 =

|Ln−1| |Ln |

 s ∈ Sn−2 l1 , l2 , l3 ∈ Ln s.t. l1 l2 l3 = e

(s, l1 , e, l2 , e, l3 , s˜ , e).

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(vi) Conditional Expectation Tangle from Pn+1 onto Pn : For n  1, let s1 ∈ Sn−1 , s2 ∈ S n−1 , m1 , m2 ∈ Ln−1 , l ∈ Ln , h ∈ H such that μ(s1 , m1 m2 , s2 , h) = e. Then,

 = δl,e

|Ln | (s1 , m1 m2 , s2 , h) |Ln−1|

and for n = 0,

 = δh,e δk,e

|K| e. |H |

(vii) Conditional Expectation Tangle from Pn onto P1,n : For n  2 let k1 , k2 ∈ K, t ∈ T2n−3 , h ∈ H such that μ(k1 , t, k2 , h) = e. Then,

 = δh,e

|H | |K|

 k ,k ∈K s.t. k k =k2 k1

and for n = 1,

 = δh,e δk,e

|H | e. |K|

(k , t, k , e)

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4. Iterated basic construction and higher relative commutants of group-type subfactors Let P be a II1 factor and G a countable discrete group with an outer action on P , and suppose G is generated by two of its finite subgroups H and K. Consider the associated group-type subfactor P H ⊂ P  K. In this section we will give a concrete realization of the Jones tower associated to this subfactor and compute its higher relative commutants. See also [19] for related results. First, let us recall the following characterization of the basic construction of a finite index subfactor [15]. Lemma 4.1. Let N ⊂ M be a finite index subfactor with E : M → N being the trace-preserving conditional expectation, B be a II1 factor containing M as a subfactor and f be a projection in B satisfying: (i) f xf = E(x)f for all x ∈ M, (ii) B is the algebra generated by M and f . Then, B is isomorphic to the basic construction M1 of N ⊂ M. Second, we recall some basic facts and notations for the crossed product construction. Unless otherwise specified, we will reserve the symbol e for the identity element of a group. The crossed product P  K can be realized as the von Neumann subalgebra of L(l 2 (K) ⊗ L2 (P )) (∼ = MK (C) ⊗ L(L2 (P ))) generated by the images of P and K in the following way: 

 P  x → Ek,k ⊗ k −1 (x) ∈ MK (C) ⊗ L L2 (P ) , (4.1) k∈K

 K  k → λk ⊗ 1 ∈ MK (C) ⊗ L L2 (P )

(4.2)

where we set the convention of considering k(x) as the element of P obtained by applying the automorphism k on x (in P ) and λk is the matrix in MK (C) corresponding to left multiplication by k. Consequently, the following commutation relation holds in P  K: kxk −1 = k(x)

for all x ∈ P , k ∈ K.

(4.3)

However, P  K can also be viewed as the vector space generated by elements of the form  x k, k∈K k xk ∈ P , where the multiplication structure is given by the relation (4.3). The unique trace on P  K is given by   tr xk k = tr(xe ) k∈K

and the unique trace-preserving conditional expectation is given by   P K EP xk k = xe . k∈K

If P K denotes the fixed point subalgebra of P , then P K is isomorphic to the basic construction of P K ⊂ P where the Jones projection is given by

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e1 =

1  k∈P K |K| k∈K

implementing the conditional expectation: 1  k(x) ∈ P K |K|

EPP K (x) =

for all x ∈ P .

k∈K

The basic construction of P ⊂ P  K is isomorphic to MK (C) ⊗ P where the inclusion P  K → MK (C) ⊗ P is given by the maps (4.1), (4.2), and the corresponding Jones projection is given by e2 = Ee,e ⊗ 1 ∈ MK (C) ⊗ P implementing the conditional expectation EPP K . The next element in the tower of basic construction is given by MK (C) ⊗ (P  K) where the inclusion MK (C) ⊗ P → MK (C) ⊗ (P  K) is induced by the inclusion P ⊂ P  K and the Jones projection is given by e3 =

1  ρk −1 ⊗ k ∈ MK (C) ⊗ (P  K) |K|

(4.4)

k∈K

implementing the conditional expectation: M (C)⊗P

K EP K

(Ek1 ,k2 ⊗ x) =

1 k1 xk2 −1 ∈ P  K |K|

for all x ∈ P , k1 , k2 ∈ K,

(4.5)

where ρk is the matrix in MK (C) corresponding to right multiplication by k. Coming back to the context of group-type subfactors we consider the unital inclusions P H → P  K → MK (C) ⊗ (P  H ) where the second inclusion factors through MK (C) ⊗ P in the obvious way. Lemma 4.2. MK (C) ⊗ (P  H ) is the basic construction for P H ⊂ P  K with Jones projection 1 e1 = Ee,e ⊗ |H | h∈H h. Proof. We need to  show that conditions (i) and (ii) of Lemma 4.1 are satisfied. To show (i), let us assume that x˜ = k∈K xk k denotes a typical element of P  K, 

   1  1  ˜ 1 = Ee,e ⊗ h x˜ Ee,e ⊗ h e1 xe |H | |H | h∈H h ∈H      1  1  −1 = Ee,e ⊗ h Ek ,k λk ⊗ k (xk ) Ee,e ⊗ h |H | |H | =

Ee,e λk Ee,e ⊗

k∈K

 = Ee,e ⊗

1 |H |2

 h∈H h ∈H

h ∈H

k ∈K k∈K

h∈H



1 |H |2

 h∈H h ∈H

h(xe )hh



hxk h



D. Bisch et al. / Journal of Functional Analysis 257 (2009) 20–46

37

whereas    1  −1 1  Ek,k ⊗ k h(xe ) Ee,e ⊗ h E(x)e ˜ 1= |H | |H | k∈K h∈H h ∈H   1  . h(x )h = Ee,e ⊗ e |H |2 h∈H 

h ∈H

Therefore, LHS = RHS. To show (ii), it is enough to show that elements of the form Ek1 ,k2 ⊗ xh for x ∈ P , h ∈ H , Let us denote the Jones projection in P  H k1 , k2 ∈ K are in the algebraic span of P  K and e1 . corresponding to the inclusion P H ⊂ P by f = |H1 | h∈H h. Thus, e1 = Ee,e ⊗ f ∈ MK (C) ⊗ P  H. This implies P e1 P = Ee,e ⊗ Pf P = Ee,e ⊗ P  H ⊂ MK (C) ⊗ P  H where P in the left-hand side is identified with its image inside MK (C) ⊗ P  H (namely, the prescription given by (4.1)). Thus the algebraic span of P  K and e1 contains elements of the type Ee,e ⊗ xh for x ∈ P , h ∈ H . To obtain elements of the form Ek1 ,k2 ⊗ xh, note that the relation λk1 Ee,e λk −1 = Ek1 ,k2 holds in MK (C). 2 2

Lemma 4.3. MK (C) ⊗ MH (C) ⊗ (P  K) is the basic construction for P  K ⊂ MK (C) ⊗ 1 (P  H ) where the Jones projection is given by e2 = |K| k∈K ρk −1 ⊗ Ee,e ⊗ k. M (C)⊗(P H )

K Proof. The conditional expectation EP K

M (C)⊗P

K EP K

is the composition

M (C)⊗(P H )

◦ EMKK (C)⊗P

.

1 k1 xk2 −1 . Therefore, E(Ek1 ,k2 ⊗ xh) = δh,e |K| To show condition (i) of Lemma 4.1,

e2 (Ek1 ,k2 ⊗ xh)e2 =

1



ρk −1 ⊗ Ee,e ⊗ k

 

|K|2 k ∈K   ρk −1 ⊗ Ee,e ⊗ k ×

 Ek1 ,k2 ⊗ Eh ,h λh ⊗ h −1 (x)

h ∈H

k ∈K

= δh,e = δh,e

1 |K|2 1 |K|2

 



ρk −1 Ek1 ,k2 ρk −1 ⊗ Ee,e λh Ee,e ⊗ k xk

k ,k ∈K

 

k ,k ∈K



Ek1 k −1 ,k2 k ⊗ Ee,e ⊗ k xk









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whereas E(Ek1 ,k2 ⊗ xh)e2    

−1  1 −1 k (x) = δh,e λk1 Ek,k λk −1 ⊗ Eh ,h ⊗ h ρk −1 ⊗ Ee,e ⊗ k 2 |K|2 h ∈H k∈K k ∈K   1 −1 λk1 Ek,k λk −1 ρk −1 ⊗ Ee,e ⊗ k (x)k = δh,e 2 |K|2 k∈K    1 −1 Ek1 k,k2 kk ⊗ Ee,e ⊗ k xkk = δh,e |K|2 k,k ∈K and the two sides are the same after renaming the indices. To show condition (ii), note that MK (C) ⊗ (P  K) is algebraically generated by MK (C) ⊗ P 1  and |K| k∈K ρk −1 ⊗ k (by the remarks preceding Eq. (4.4)). The following holds in MK (C) ⊗ sits inside MH (C) ⊗ P (since in the MH (C) ⊗ (P  K) because of this fact and the way P second tensor component we get expressions of the form h,h ∈H Eh,h Ee,e Eh ,h which reduces to Ee,e ):

  MK (C) ⊗ P e2 MK (C) ⊗ P = MK (C) ⊗ Ee,e ⊗ (P  K)  

⇒ MK (C) ⊗ P  H e2 MK (C) ⊗ P  H = MK (C) ⊗ MH (C) ⊗ (P  K) where the last implication is again due to the relation λh1 Ee,e λh−1 = Eh1 ,h2 in MH (C). 2

2

Thus, we have the first two levels in the tower of basic construction: P H ⊂ P  K ⊂ MK (C) ⊗ (P  H ) ⊂ MK×H (C) ⊗ (P  K) where we identify MK×H (C) with MK (C) ⊗ MH (C). The next levels in the tower are obvious generalizations and we gather everything in the following proposition. Proposition 4.4. Let G be a group acting outerly on the II1 factor P and assume G is generated by two of its finite subgroups H and K. Then the nth element of the tower of basic construction of the group-type subfactor N = P H ⊂ P  K = M is given by Mn ∼ = MSn (C) ⊗ (P  Ln ) where the inclusion of Mn inside Mn+1 is as follows: Mn  Es,t ⊗ x →



Es,t ⊗ El,l ⊗ l −1 (x) ∈ Mn+1

for all x ∈ P , s, t ∈ Sn ,

l∈Ln

Mn  Es,t ⊗ l → Es,t ⊗ λl ⊗ e ∈ Mn+1

for all l ∈ Ln , s, t ∈ Sn ,

D. Bisch et al. / Journal of Functional Analysis 257 (2009) 20–46

and the nth Jones projection is:  1  Mn  e n =

l∈Ln IMSn−2 ⊗ ρl −1 |Ln | 1  h∈H Ee,e ⊗ h, |H |

39

⊗ Ee,e ⊗ l, if n > 1, if n = 1.

Proof. We use induction. The case of n = 1 is a little different from the rest and is proved in Lemma 4.2 and the n = 2 case is proved in Lemma 4.3. Suppose the statement of the above proposition holds upto a level n (> 2). Now, the subfactor Mn−1 ⊂ Mn is isomorphic to: MSn−1 ⊗ P  Ln−1 ⊂ MSn−1 ⊗ MLn−1 ⊗ P  Ln where we identify MSn−1 ⊗ MLn−1 with MSn and the inclusion is induced by identity over MSn−1 tensored with the inclusion of the subfactor P  Ln−1 ⊂ MLn−1 ⊗ P  Ln . Using Lemma 4.3 for K = Ln−1 and H = Ln , it is clear that the statement of the proposition holds for level n + 1. 2 Remark 4.5. The formula for the unique trace-preserving conditional expectation is: n EM Mn−1 (Es1 ,s2 ⊗ Em1 ,m2 ⊗ xl) = δl,e

1 Es ,s ⊗ m1 xm−1 2 |Ln | 1 2

where s1 , s2 ∈ Sn−1 , m1 , m2 ∈ Ln−1 , l ∈ Ln and x ∈ P and the unique trace on Mn is given by trMn (Es1 ,s2 ⊗ xl) =

1 δl,e δs1 ,s2 trM (x) |Sn |

where s1 , s2 ∈ Sn , l ∈ Ln and x ∈ P . We will now compute the higher relative commutants using the above model of the Jones tower. To this end, we need the following two lemmas, where we denote the set of automorphisms of M ⊃ N that fixes elements of N pointwise by Gal(N ⊂ M). Lemma 4.6. Let N ⊂ M be a finite index subfactor and θ ∈ Gal(N ⊂ M), then the bimodule 2 2 M L (θ )M (where the module is L (M) with usual left action of M but right action is twisted by θ ) is a 1-dimensional irreducible sub-bimodule of M L2 (M1 )M . Proof. Define uθ : L2 (M) → L2 (M) by uθ (xΩ) = θ (x)Ω for x ∈ M where Ω is the cyclic and separating vector in the GNS construction with respect to the canonical trace tr. Note that uθ (n1 · xΩ · n2 ) = uθ (n1 xn2 Ω) = n1 · θ (x)Ω · n2 for n1 , n2 ∈ N , x ∈ M. This implies uθ ∈ N ∩ M1 . Now define T : M L2 (θ )M → M L2 (M1 )M by T (xΩ) = xuθ Ω1 for x ∈ M. It is completely routine to check that T is a well-defined M-M linear isometry and we leave this to the reader. 2 Corollary 4.7. H = Gal(P H ⊂ P ). Proof. Clearly H ⊂ Gal(P H ⊂ P ). Let θ ∈ Gal(P H ⊂ P ). Note that P L2 (P  H )P ∼ = 2 (h) . Thus by Lemma 4.6, L2 (θ ) ∼ L2 (h) for some h ∈ H . This implies L = P P P P P h∈H P θ h−1 ∈ Inn(P ) ∩ Gal(P H ⊂ P ) = {idP } since P H ⊂ P is irreducible. Hence, θ ∈ H . 2

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Lemma 4.8. Let N ⊂ M be an irreducible subfactor, i.e. N ∩ M ∼ = C and θ ∈ Aut(M). For x ∈ M, the following are equivalent: (i) x = 0 and xθ (y) = yx for all y ∈ N , x ∈ U(M) and Adx0 ◦ θ ∈ Gal(N ⊂ M). (ii) x0 := x Proof. (ii) ⇒ (i) part is easy. For (i) ⇒ (ii), note that we also have θ (y)x ∗ = x ∗ y for all y ∈ N . Thus, x ∗ x ∈ θ (N ) ∩ M and ∗ xx ∈ N ∩ M. Since N ∩ M ∼ = C, xx ∗ = x ∗ x = x2 . Hence, x0 ∈ U(M) and x0 θ (y)x0 ∗ = y for all y ∈ N . This implies Adx0 ◦ θ ∈ Gal(N ⊂ M). 2 Proposition 4.9. For the group-type subfactor N = P H ⊂ P  K = M, the relative commutants N ∩ Mn and M ∩ Mn are given by ⎧ C, ⎧ ⎪  ⎫ if n = −1, ⎪ ⎨  s , s ∈ S ⎨ ⎬ 1 2 n  N ∩ Mn ∼ = span E  l ∈ Ln ⊗ l , if n  0, ⎪ s ,s  ⎪ ⎩ 1 2 ⎩  μ(s1 )lμ(s2 )−1 ∈ H ⎭ ⎧ C, ⎧ ⎪  ⎫ if n = 0, ⎪ ⎨  t , t ∈ T ⎨ ⎬ n−1  1 2 M ∩ Mn ∼ = span  Ek,kk0 ⊗ Et1 ,t2 ⊗ l  μ(t1 )lμ(t2 )−1 ∈ K , if n  1. ⎪ k∈K ⎪ ⎩ ⎩  k0 = μ(t1 )lμ(t2 )−1 ⎭ Proof. We compute the relative commutants in relation to the concrete model of the basic construction described in Proposition 4.4. Consider the inclusion  Es,s ⊗ μ(s)−1 (x) ∈ MSn (C) ⊗ (P  Ln ) = Mn . N = P H  x → s∈Sn

Let w=

 s1 ,s2 ∈Sn l∈Ln

Es1 ,s2 ⊗ xsl1 ,s2 l ∈ N ∩ Mn ,

for some xsl1 ,s2 ∈ P .

Then, wy = yw ⇔

for all y ∈ N 

 Es1 ,s2 ⊗ xsl1 ,s2 l μ(s2 )−1 (y)

s1 ,s2 ∈Sn l∈Ln

=



s1 ,s2 ∈Sn l∈Ln

⇔ ⇔

Es1 ,s2 ⊗ μ(s1 )−1 (y)xsl1 ,s2 l

for all y ∈ N

 xsl1 ,s2 lμ(s2 )−1 (y) = μ(s1 )−1 (y)xsl1 ,s2 for all y ∈ N, s1 , s2 ∈ Sn , l ∈ Ln   

for all y ∈ N, s1 , s2 ∈ Sn , l ∈ Ln . μ(s1 ) xsl1 ,s2 μ(s1 )lμ(s2 )−1 (y) = yμ(s1 ) xsl1 ,s2

D. Bisch et al. / Journal of Functional Analysis 257 (2009) 20–46

41

Now, by Lemma 4.8 and Corollary 4.7, for s1 , s2 ∈ Sn and l ∈ Ln , xsl1 ,s2

= 0



Adx0 ◦ μ(s1 )lμ(s2 )−1 ∈ H

⇔

where x0 =

μ(s1 )(xsl1 ,s2 ) μ(s1 )(xsl1 ,s2 )

.

Moreover, xsl1 ,s2 = 0 ⇒ Adx0 ∈ G ∩ Inn(P ) = {idP }. Since P is a factor, x0 ∈ C1. Thus,

xsl1 ,s2 will be non-zero only if μ(s1 )lμ(s2 )−1 ∈ H and in such cases xsl1 ,s2 will be a scalar multiple of identity. Hence, N ∩ Mn is spanned by the linearly independent set {Es1 ,s2 ⊗ l: s1 , s2 ∈ Sn , l ∈ Ln , μ(s1 )lμ(s2 )−1 ∈ H }. For M ∩ Mn where n  1, we consider the inclusion M = P  K ⊃ P  x →



Es,s ⊗ μ(s)−1 (x) ∈ MSn (C) ⊗ (P  Ln ) = Mn ,

s∈Sn

M = P  K ⊃ K  k → λk ⊗ ITn−1 ⊗ 1 ∈ MSn (C) ⊗ (P  Ln ) = Mn . Let w=

 s1 ,s2 ∈Sn l∈Ln

Es1 ,s2 ⊗ xsl1 ,s2 l ∈ M ∩ Mn ,

for some xsl1 ,s2 ∈ P .

Using arguments similar to those for the calculations for the case of N ∩ Mn it follows that wy = yw ⇔

for all y ∈ P   

μ(s1 ) xsl1 ,s2 μ(s1 )lμ(s2 )−1 (y) = yμ(s1 ) xsl1 ,s2

for all y ∈ P , s1 , s2 ∈ Sn , l ∈ Ln ,

and hence xsl1 ,s2 = 0 ⇔ μ(s1 )lμ(s2 )−1 = e since P is a factor; further, in such cases, xsl1 ,s2 is a scalar multiple of 1. Now, wk = kw ⇔ ⇔

for all k ∈ K

k −1 wk = w for all k ∈ K    l λ−1 k ⊗ ITn−1 ⊗ 1 Es1 ,s2 ⊗ xs1 ,s2 l (λk ⊗ ITn−1 ⊗ 1) s1 ,s2 ∈Sn l∈Ln

=



s1 ,s2 ∈Sn l∈Ln

⇔



k1 , k2 ∈ K t1 , t2 ∈ Tn−1 l ∈ Ln

Es1 ,s2 ⊗ xsl1 ,s2 l

for all k ∈ K

l λ−1 k Ek1 ,k2 λk ⊗ Et1 ,t2 ⊗ x(k1 ,t1 ),(k2 ,t2 ) l

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D. Bisch et al. / Journal of Functional Analysis 257 (2009) 20–46



=

l Ek1 ,k2 ⊗ Et1 ,t2 ⊗ x(k l 1 ,t1 ),(k2 ,t2 )

for all k ∈ K

k1 , k2 ∈ K t1 , t2 ∈ Tn−1 l ∈ Ln



⇔

l Ek −1 k1 ,k −1 k2 ⊗ Et1 ,t2 ⊗ x(k l 1 ,t1 ),(k2 ,t2 )

k1 , k2 ∈ K t1 , t2 ∈ Tn−1 l ∈ Ln

=



l Ek1 ,k2 ⊗ Et1 ,t2 ⊗ x(k l 1 ,t1 ),(k2 ,t2 )

for all k ∈ K

k1 , k2 ∈ K t1 , t2 ∈ Tn−1 l ∈ Ln

l l = x(kk x(k 1 ,t1 ),(k2 ,t2 ) 1 ,t1 ),(kk2 ,t2 )

⇔

for all k, k1 , k2 ∈ K, t1 , t2 ∈ Tn−1 , l ∈ Ln .

Finally, combining the conditions that we get from considering commutation of w with elements of P and K, we can express w as a linear combination of elements of the form: 

Ekk1 ,kk2 ⊗ Et1 ,t2 ⊗ l

k∈K

where k1 , k2 ∈ K, t1 , t2 ∈ Tn−1 , l ∈ Ln such that k1 μ(t1 )lμ(t2 )−1 k2−1 = e. Equivalently, w can be realized as a linear combination of  k∈K

Ekk −1 ,k ⊗ Et1 ,t2 ⊗ l = 0



Ek,kk0 ⊗ Et1 ,t2 ⊗ l

k∈K

where t1 , t2 ∈ Tn−1 , l ∈ Ln such that μ(t1 )lμ(t2 )−1 ∈ K and k0 = μ(t1 )lμ(t2 )−1 .

2

Remark 4.10. (i) The set  ⎫  s1 , s2 ∈ Sn , ⎬  Es1 ,s2 ⊗ l  l ∈ Ln , ⎩  μ(s1 )lμ(s2 )−1 ∈ H ⎭ ⎧ ⎨

 ⎫#  t1 , t2 ∈ Tn−1 , ⎬  Ek,kk0 ⊗ Et1 ,t2 ⊗ l  μ(t1 )lμ(t2 )−1 ∈ K, resp. ⎩  k0 = μ(t1 )lμ(t2 )−1 ⎭ k∈K

"

⎧ ⎨

forms a basis of N ∩ Mn (resp. M ∩ Mn ). (ii) The unique trace-preserving conditional expectation from N ∩ Mn onto M ∩ Mn is given by

∩Mn EN M ∩Mn (Ek1 ,k2 ⊗ Et1 ,t2 ⊗ l) = δk1 μ(t1 )l,k2 μ(t2 )

1  Ek,kk1 −1 k2 ⊗ Et1 ,t2 ⊗ l |K| k∈K

where k1 , k2 ∈ K, t1 , t2 ∈ Tn−1 , l ∈ Ln such that k1 μ(t1 )lμ(t2 )−1 k2−1 ∈ H . To see (ii), we need to check that for t1 , t2 ∈ Tn−1 and k0 = μ(t1 )l μ(t2 )−1 ∈ K,

D. Bisch et al. / Journal of Functional Analysis 257 (2009) 20–46

tr

$  k ∈K

 % Ek ,k k0 ⊗ Et1 ,t2 ⊗ l (Ek1 ,k2 ⊗ Et1 ,t2 ⊗ l)

= δl l,e δt2 ,t1 δt1 ,t2 δk2 k0 ,k1

1 |Sn |

= δk1 μ(t1 )l,k2 μ(t2 ) δl l,e δt1 ,t2 δt2 ,t1 δe,k0 k1 −1 k2 = tr

43

$  k ∈K

Ek ,k k0 ⊗ Et1 ,t2 ⊗ l





1 |Sn |

% 1  δk1 μ(t1 )l,k2 μ(t2 ) Ek,kk1 −1 k2 ⊗ Et1 ,t2 ⊗ l |K| k∈K

which is a routine computation using the second part of Remark 4.5. 5. The planar algebra of group-type subfactors In this section, our main goal is to show that the planar algebra defined in Section 3, is isomorphic to the one arising from the group-type subfactor P H ⊂ P  K. Conversely, we will show that any subfactor whose standard invariant is given by such planar algebra, is indeed of this type. We will use the following well-known fact regarding isomorphism of two planar algebras and Theorem 4.2.1 of [11] describing the planar algebra arising from an extremal finite index subfactor. Fact. Let P 1 and P 2 be two planar algebras. Then P 1 ∼ = P 2 if and only if there exists a vector 1 2 space isomorphism ψ : P → P such that: (i) ψ preserves the filtered algebra structure, (ii) ψ preserves the actions of all possible Jones projection tangles and the (two types of) conditional expectation tangles. If P1 and P2 are ∗-planar algebras, then we consider such ψ that are ∗-preserving to be a ∗-planar algebra isomorphism. Let us denote the planar algebra in Section 3 by P BH and the one arising from the group-type subfactor N = P H ⊂ P  K = M by P N ⊂M . Theorem 5.1. P N ⊂M ∼ = P BH . Proof. By Theorem 4.2.1 of [11] we have that PnN ⊂M = N ∩ Mn where the nth Jones projection N ∩M

N ⊂M is δEN ∩Mn+1 tangle is given by δen , the conditional expectation tangle from PnN ⊂M onto Pn−1 n N ∩M

N ⊂M and the conditional expectation tangle from PnN ⊂M onto P1,n is δEM ∩Mn+1 . n+1 Define the map

ψ : P N ⊂M → P BH ∪

∪

by ψn (Es1 ,s2 ⊗ l) = (s1 , l, s˜2 , h)

ψn : PnN ⊂M → PnBH where n  0, s1 , s2 ∈ Sn , l ∈ Ln such that μ(s1 )lμ(s2 )−1 ∈ H and μ(s1 )lμ(s2 )−1 h = e.

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Clearly, ψ is a vector space isomorphism by definition. In order to check that ψ is a filtered ∗-algebra isomorphism, we use Remark 3.3(i), (ii), (iii) and (iv). For instance, to show that ψn is an algebra homomorphism, we need to show ψn (Es1 ,s2 ⊗ l1 · Es3 ,s4 ⊗ l2 ) = ψn (Es1 ,s2 ⊗ l1 ) · ψn (Es3 ,s4 ⊗ l2 )

 ⇐ δs2 ,s3 s1 , l1 l2 , s˜4 , μ(s4 )l2−1 l1−1 μ(s1 )−1

  = s1 , l1 , s˜2 , μ(s2 )l1−1 μ(s1 )−1 · s3 , l2 , s˜4 , μ(s4 )l2−1 μ(s3 )−1 which indeed holds by Remark 3.3(iii). Now, it remains to be shown that ψ preserves the action of Jones projection tangles and the two types of conditional expectation tangles. For this, we use Remark 3.3(v), (vi) and (vii). Proof of each of the three kinds of tangles is completely routine; however, we will discuss the action of conditional expectation tangle in details. Let us consider the conditional expectation tangle from n + 1 (color of the internal disc) to color n (color of the external disc) applied to the element N ⊂M Es1 ,s2 ⊗ Em1 ,m2 ⊗ l ∈ Pn+1 = N ∩ Mn ; by Jones’s theorem (4.2.1 of [11]) and Remark 4.5, the output should be 

|Ln | Es ,s ⊗ m1 m−1 2 |Ln+1| 1 2   |Ln | ψn s1 , m1 m−1 −→ δl,e , s˜2 , μ(s2 )m2 m−1 μ(s1 )−1 . 2 1 |Ln+1|

δl,e

On the other hand, by Remark 3.3(vi), the conditional expectation tangle applied to −1 −1 ψn+1 (Es1 ,s2 ⊗ Em1 ,m2 ⊗ l) = (s1 , m1 , l −1 , m−1 2 , s˜2 , μ(s2 )m2 lm1 μ(s1 ) ) is given by & −1 −1 n| δl,e |L|Ln+1| (s1 , m1 m−1 2 , s˜2 , μ(s2 )m2 m1 μ(s1 ) ). This completes the proof.

2

Corollary 5.2. Given any countable discrete group G generated by two of its finite subgroups, there exists a hyperfinite subfactor with standard invariant described by P BH . Moreover, P BH is a spherical C ∗ planar algebra. Proof. The proof follows from the fact that any countable discrete group G has an outer action on the hyperfinite II1 factor. 2 Theorem 5.3. Given a subfactor N ⊂ M with standard invariant isomorphic to P BH , there exists an intermediate subfactor N ⊂ P ⊂ M and outer actions of H and K on P such that (N ⊂ M)  (P H ⊂ P  K). Proof. Let P N ⊂M denote the planar algebra of N ⊂ M formed by its relative commutants and φ : P N ⊂M → P BH be a planar algebra isomorphism. Consider the element q = φ −1 (e, e, e, e) ∈ N ∩ M1 . Clearly, (i) q is a projection, (ii) qe1 = e1 , and (iii) EM (q) = |K|−1 1. Using action of tangles and the planar algebra isomorphism φ, it also follows that

D. Bisch et al. / Journal of Functional Analysis 257 (2009) 20–46

45

(iv)

The conditions (i)–(iv) assert that q is an intermediate subfactor projection as described in [2]. Define P = M ∩ {q} . To show P is a factor, first note that

   P ∩ P ⊂ N ∩ P = N ∩ M ∩ {q} = φ P1BH ∩ (e, e, e, e) . So, it is enough to show that P1BH ∩ {(e, e, e, e)} = C1. If x = {(e, e, e, e)} , then (e, e, e, e) · x = x · (e, e, e, e)

⇒





g∈K∩H (g, g

−1 )

∈ P1BH ∩





 λg e, e, g, g −1 = λg g, e, e, g −1 .

g∈K∩H

g∈K∩H

Hence, λg = δg,e λe for all g ∈ K ∩ H and x ∈ C1. It remains to establish that N (resp. M) is the fixed-point subalgebra (resp. crossed-product algebra) of P with respect to an outer action of the group H (resp. K). It is easy to prove (see [11]) that if the standard invariant of a subfactor N˜ ⊂ M˜ is given by the planar algebra corresponding to the fixed-point subfactor (resp. crossed-product subfactor) with respect to a ˜ then there exists an outer action of G ˜ on M˜ (resp. N˜ ) such that N˜ (resp. M) ˜ finite group G, ˜ is isomorphic to fixed-point subalgebra (crossed-product algebra) of the action. Again, if N ⊂ P˜ ⊂ M˜ is an intermediate subfactor and q˜ is its corresponding intermediate subfactor projection, ˜ is given by the range of the idempotent tangle then the planar algebra of N˜ ⊂ P˜ (resp. P˜ ⊂ M)

or

(resp.

according as n is even or odd ([6], see also [1]).

or

)

46

D. Bisch et al. / Journal of Functional Analysis 257 (2009) 20–46

Getting back to our context, to get the planar algebra of N ⊂ P (resp. P ⊂ M), the elements in the image of the above idempotent tangles are given by the words with letters coming from K and H alternately where every element coming from K (respectively H ) must necessarily be e. Such a planar algebra is the same as P BH with K = {e} (resp. H = {e}); by Theorem 5.1 this is indeed the planar algebra corresponding to fixed-point subfactor (resp. crossed-product subfactor) with respect to H (resp. K). 2 References [1] B. Bhattacharyya, Z. Landau Intermediate standard invariants and intermediate planar algebras, preprint. [2] D. Bisch, A note on intermediate subfactors, Pacific J. Math. 163 (2) (1994) 201–216. [3] D. Bisch, Bimodules, higher relative commutants and the fusion algebra associated to a subfactor, in: Operator Algebras and Their Applications, in: Fields Inst. Commun., vol. 13, Amer. Math. Soc., Providence, RI, 1997, pp. 13– 63. [4] D. Bisch, U. Haagerup, Composition of subfactors: New examples of infinite depth subfactors, Ann. Sci. École Norm. Sup. 29 (1996) 329–383. [5] D. Bisch, V. Jones, Algebras associated to intermediate subfactors, Invent. Math. 128 (1) (1997) 89–157. [6] D. Bisch, V. Jones, The free product of planar algebras, and subfactors, in preparation. [7] D. Bisch, S. Popa, Examples of subfactors with Property (T) standard invariant, Geom. Funct. Anal. 9 (1999) 215– 225. [8] D. Evans, Y. Kawahigashi, Quantum Symmetries on Operator Algebras, Oxford Math. Monogr., Oxford Science Publications, Oxford University Press, New York, 1998, xvi+829 pp. [9] F. Goodman, P. de la Harpe, V. Jones, Coxeter Graphs and Towers of Algebras, Math. Sci. Res. Inst. Publ., vol. 14, Springer-Verlag, New York, 1989, x+288 pp. [10] V. Jones, Index for subfactors, Invent. Math. 72 (1983) 1–25. [11] V. Jones, Planar algebras, arxiv:math/9909027v1. [12] V. Jones, Two subfactors and the algebraic decomposition of bimodules over II1 factors, preprint. [13] V. Jones, V.S. Sunder, Introduction to Subfactors, London Math. Soc. Lecture Note Ser., vol. 234, ISBN 0-52158420-5, 1997. [14] A. Ocneanu, Quantized groups, string algebras and Galois theory for algebras, in: Operator Algebras and Applications, vol. 2, in: London Math. Soc. Lecture Note Ser., vol. 136, Cambridge Univ. Press, Cambridge, 1988, pp. 119–172. [15] M. Pimsner, S. Popa, Iterating the basic construction, Trans. Amer. Math. Soc. 310 (1) (1988) 127–133. [16] S. Popa, Classification of amenable subfactors of type II, Acta Math. 172 (1994) 163–255. [17] S. Popa, An axiomatization of the lattice of higher relative commutants of a subfactor, Invent. Math. 120 (1995) 427–445. [18] S. Popa, Some properties of the symmetric enveloping algebra of a subfactor, with applications to amenability and property T, Doc. Math. 4 (1999) 665–744. [19] J.-M. Vallin, Relative matched pairs of finite groups from depth two inclusions of von Neumann algebras to quantum groupoids, J. Funct. Anal. 254 (2) (2008) 2040–2068.

Journal of Functional Analysis 257 (2009) 47–87 www.elsevier.com/locate/jfa

Noncommutative ball maps J. William Helton a,1 , Igor Klep b,c,2 , Scott McCullough d,∗,3 , Nick Slinglend a a Department of Mathematics, University of California, San Diego, United States b Univerza v Ljubljani, Fakulteta za matematiko in fiziko, Slovenia c Univerza v Mariboru, Fakulteta za naravoslovje in matematiko, Slovenia d Department of Mathematics, University of Florida, Gainesville, FL, United States

Received 8 October 2008; accepted 10 March 2009

Communicated by N. Kalton

Abstract In this paper, we analyze problems involving matrix variables for which we use a noncommutative algebra setting. To be more specific, we use a class of functions (called NC analytic functions) defined by power series in noncommuting variables and evaluate these functions on sets of matrices of all dimensions; we call such situations dimension-free. These types of functions have recently been used in the study of dimension-free linear system engineering problems. In this paper we characterize NC analytic maps that send dimension-free matrix balls to dimension-free matrix balls and carry the boundary to the boundary; such maps we call “NC ball maps”. We find that up to normalization, an NC ball map is the direct sum of the identity map with an NC analytic map of the ball into the ball. That is, “NC ball maps” are very simple, in contrast to the classical result of D’Angelo on such analytic maps in C. Another mathematically natural class of maps carries a variant of the noncommutative distinguished boundary to the boundary, but on these our results are limited. We shall be interested in several types of noncommutative balls, conventional ones, but also balls defined by constraints called Linear Matrix Inequalities (LMI). What we do here is a small piece of the bigger puzzle of understanding how LMIs behave with respect to noncommutative change of variables. © 2009 Elsevier Inc. All rights reserved.

* Corresponding author.

E-mail addresses: [email protected] (J.W. Helton), [email protected] (I. Klep), [email protected] (S. McCullough), [email protected] (N. Slinglend). 1 Research supported by NSF grants DMS-0700758, DMS-0757212, and the Ford Motor Co. 2 Supported by the Slovenian Research Agency (project No. Z1-9570-0101-06). 3 Research supported by the NSF grant DMS-0758306. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.03.008

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Keywords: Noncommutative analytic function; Complete isometry; Ball map; Linear matrix inequality

1. Introduction In the introduction we will state some of our main results. For this we need to start with the definitions of NC polynomials (Section 1.1) and NC analytic maps (Section 1.2). We then proceed to define NC ball maps in Section 1.3, where we explain what it means for an NC ball map to map ball to ball with boundary to boundary. After that we can and do state our main results classifying NC ball maps in Sections 1.3 and 1.4. Finally, the introduction concludes by considering two types of generalizations, the first being to balls defined by LMIs, the second being to NC analytic maps carrying special sets on the boundary of a ball to the boundary of a ball. Before continuing with the more detailed introduction, we pause to offer some perspective and mention related significant contributions. Ball maps form a distinguished subset of the space of noncommutative analytic functions on a noncommutative domain and we direct the interested reader to the elegant general theory of noncommutative analytic functions developed in the articles [12–14,29,30,16,17] for a more complete account than found here and to the work of Popescu on free analytic functions. Some of these references are [22,27,23,24,26]. Noncommutative rational, Schur class and analytic functions are also intimately related to systems theory. A small sample of the references includes [2,3,10,12,18]. The noncommutative balls that we consider are modeled on g  × g matrices, and in the special case that g = 1, they correspond to those studied by Popescu for operator, not just matrix, coefficients. Precomposition by an automorphism of the domain preserves ball maps and such automorphisms are studied at various levels of generality in [28,17]. Linear ball maps are an important special case, and these are identified with completely isometric mappings from one operator space into another. The books [19,20,6] provide comprehensive introductions to the theory of operator systems, spaces, and algebras, and the papers [5] and [4] treat very generally complete isometries into a C-star algebra. 1.1. Words and NC polynomials Let g  , g ∈ N. We write x for the monoid freely generated by x, i.e., x consists of words in the g  g letters x11 , . . . , x1g , x21 , . . . , xg  g (including the empty word ∅ which plays the role of the identity 1). Let Cx denote the associative C-algebra freely generated by x, i.e., the elements of Cx are polynomials in the noncommuting variables x with coefficients in C. Its elements are called NC polynomials. An element of the form aw where 0 = a ∈ C and w ∈ x is called a monomial and a its coefficient. Hence words are monomials whose coefficient is 1. Let ∗ , . . . , x ∗ ) denote another set of g  g symbols. We shall also consider the free algebra x ∗ = (x11 g g Cx, x ∗  that comes equipped with the natural involution xij → xij∗ . For example,   ∗ 2  2 ∗ ∗ ∗ 1 + ix11 x23 x34 = 1 − ix34 x23 x11 . (Here i denotes the imaginary unit

√

−1.)

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49

1.1.1. NC matrix polynomials A matrix-valued NC polynomial is an NC polynomial with matrix coefficients. We shall use the phrase scalar NC polynomial if we want to emphasize the absence of matrix constructions. Often when the context makes the usage clear we drop adjectives such as scalar, 1 × 1, matrix polynomial, matrix of polynomials and the like. 1.1.2. Polynomial evaluations  If p is an NC polynomial in x and X ∈ (Cn×n )g ×g , the evaluation p(X) is defined by simply replacing xij by Xij . For example, if p(x) = Ax11 x21 , where 

 −4 3 2 A= , 2 −1 0 then  p

⎡ 0 4 0 −3 0       0 1 1 0 0 1 1 0 3 0 2 ⎢ −4 0 , =A⊗ =⎣ 1 0 0 −1 1 0 0 −1 0 −2 0 1 0 2 0 −1 0 0

⎤ −2 0 ⎥ ⎦. 0 0



On the other hand, if p(x) = A and X ∈ (Cn×n )g ×g , then p(X) = A ⊗ In . The tensor product in the expressions above is the usual (Kronecker) tensor product of matrices. Thus we have reserved the tensor product notation for the tensor product of matrices and have eschewed the strong temptation of using A ⊗ xk in place of Axk when xk is one of the noncommuting variables. 1.2. Definition of NC analytic functions An elegant theory of noncommutative analytic functions is developed in the articles [12–14, 29,30]; see also [22]. What we need in this article are specializations of definitions of these papers. In this section we summarize the definitions and properties needed in the sequel. For d  , d ∈ N define Bd  ×d :=

∞    d  ×d    Idn − X ∗ X  0 , X ∈ Cn×n

(1.1)

n=1

int Bd  ×d :=

∞    d  ×d    Idn − X ∗ X  0 , X ∈ Cn×n

(1.2)

n=1

∂Bd  ×d :=

∞     d  ×d   X = 1 , X ∈ Cn×n

(1.3)

n=1

Md  ×d :=

∞   n×n d  ×d C . n=1

(1.4)

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We shall occasionally use the notation     ,d  Xj, ∈ CN ×N , X  1 , Bd  ×d (N ) = X = [ Xj, ]dj,=1     ,d  Xj, ∈ CN ×N . Md  ×d (N ) = X = [ Xj, ]dj,=1 

 g ×g is the (disjoint) Given  g , g ∈ N, the noncommutative (NC) ε-neighborhood of 0 in C  union N ∈N {X ∈ Mg ×g (N ) | X < ε}. An open NC domain D containing 0 (in its interior) is a union N DN of open sets DN ⊆ Mg  ×g (N ) which is closed with respect to direct sums and such that there is an ε > 0 such that D contains the NC ε-neighborhood of 0. A d  × d NC analytic function f on an open NC domain D containing 0 is defined as follows:

(1) f has an NC power series, for which there exists an NC ε > 0 neighborhood of 0 on which it is convergent. That is, 

f=

aw w

(1.5)

w∈x 

for aw ∈ Cd ×d and for every N ∈ N and every g  × g-tuple of square matrices X ∈ Bg  ×g with X < ε the series f (X) =



aw ⊗ w(X)

(1.6)

w∈x

converges. We interpret convergence for a given X as conditional convergence of the series ∞  

aw ⊗ w(X).

α=0 |w|=α

Thus the order of summation is over the homogeneous parts of the power series expansion. Thus with f (α) equal to the α homogeneous part in the NC power series expansion of f , the series converges for a given X provided ∞ 

f (α) (X)

(1.7)

α=0

converges. Since both aw and w(X) are matrices, the particular norm topology chosen has no influence on convergence. The radius ε of this ball of convergence (or sometimes, by abuse of notation, the ball itself) will be called the series radius. (2) If a : W → D is a matrix-valued function analytic on a domain W in CN , the composition f ◦ a is a matrix-valued analytic function on W and continuous to ∂W. Remark 1.1. Popescu [22] has a notion of free analytic function in g  variables (that is, g = 1) based upon power series expansions like that in (1.7). His definition allows for operator coefficients, but on the other hand requires convergence of the NC power series on all of int Bg  (the

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51



NC 1-neighborhood of 0 in Cg ). It turns out that for bounded NC analytic functions with matrix coefficients the two notions are the same, see Lemma 6.1. Here we have avoided extending the theory of free analytic functions to Bg  ×g , looking forward to working on more general domains in the future. 1.2.1. Properties of NC analytic functions Proposition 1.2. Let D be an NC domain containing 0. (i) The sum of two d  × d NC analytic functions on D is a d  × d NC analytic function on D. (ii) The product of two d  × d NC analytic functions on D is a d  × d NC analytic function on D. (iii) The composition of two NC analytic functions is an NC analytic function. More precisely, if f : D → D is a d1 × d1 NC analytic function, where D is an NC domain with 0 ∈ D , and h is a d2 × d2 NC analytic function on D , then h ◦ f is a d1 d2 × d1 d2 NC analytic function on D. Proof. Properties (i) and (ii) are standard and we only consider (iii). The fact that h ◦ f admits an NC power series as in (1.5) was observed e.g. in [14,29]. The composition property (2) of Section 1.2 is easily checked. 2 More is said about properties of NC analytic functions in Section 6. 1.3. NC ball maps f and their classification when f (0) = 0 A function f : int Bg  ×g → Md×d  which is NC analytic will often be called an NC analytic function on the ball Bg  ×g and denoted f : Bg  ×g → Md×d  . An NC analytic function f : Bg  ×g → Bd  ×d mapping the boundary to the boundary is called an NC ball map. The notion of f mapping boundary to boundary is a bit complicated (because of convergence issues) so requires explanation. For a given X ∈ Bg  ×g (N ), define the function fX : D → Md×d  (N ) by z → f (zX). (Here D denotes the unit disc D = {z ∈ C | |z| < 1} in the complex plane.) If   lim fX reit r1

exists, denote that limit by f (eit X). The function f maps the boundary to the boundary if whenever X = 1 and f (eit X) exists, then   it  f e X  = 1. Since f is bounded, Fatou’s theorem implies that for each X ∈ Bg  ×g the limit fX (eit ) = f (eit X) exists for almost every t. If f is an NC ball map, X ∈ ∂Bg  ×g and f (X) is defined, then a (nonzero) vector γ such that f (X)γ  = γ  is called a binding vector and this property binding.

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Our main result on NC ball maps which map 0 to 0 is: Theorem 1.3. Let h : Bg  ×g → Bd  ×d be an NC ball map with h(0) = 0. Then there exist unitaries   U : Cd → Cd and V : Cd → Cd such that   x 0 h(x) = V U ∗, (1.8) ˜ 0 h(x) ˜ where h˜ : Bg  ×g → B(d  −g  )×(d−g) is an NC analytic contraction-valued map with h(0) = 0. Conversely, every NC analytic h satisfying (1.8) for unitaries U, V and an NC analytic contraction-valued map h˜ fixing the origin, is an NC ball map Bg  ×g → Bd  ×d sending 0 to 0. The proof of the theorem is completed in Section 4. The general result is built from the linear version of the theorem, Theorem 3.3 which appears as Corollary 3.6 in [5]. See also [4]. As an illustration of Theorem 1.3 we describe a special case. For convenience, we adopt the notation Bg  for Bg  ×1 . Corollary 1.4. If h : Bg  → Bd  is an NC ball map with h(0) = 0, then h is linear and there is   a unique isometry M ∈ Cd ×g such that h = Mx. In particular, if d  < g  then no such NC ball maps exist. ˜ column is gone. Moreover, Proof. When h maps Bg  to Bd  the h(x)   I M =V∗ 0 is an isometry.

2

1.4. NC ball maps f when f (0) is not necessarily 0 In the previous section we treated NC ball maps f with f (0) = 0, an assumption we drop in this section. The strategy is to compose f with a bianalytic automorphism of an NC ball to reduce the problem to the f (0) = 0 setting. Section 1.4.1 contains information on bianalytic mappings on an NC ball, while the main results appear in Section 1.4.2. 1.4.1. Linear fractional transformations For a given d  × d scalar matrix v with v < 1, define Fv : Bd  ×d → Bd  ×d by 1/2   −1  1/2 Fv (u) := v − Id  − vv ∗ Id − v ∗ v u Id − v ∗ u .

(1.9)

Of course it must be shown that Fv actually takes values in Bd  ×d . This is done in Lemma 1.6 below. Linear fractional transformations such as Fv are common in circuit and system theory, since they are associated with energy conserving pieces of a circuit (cf. [31]). Lemma 1.5. Suppose D is an open NC domain containing 0. If u : D → Bd  ×d is NC analytic, then Fv (u(x)) is an NC analytic function (in x) on D. Proof. See Section 5.

2

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53

Notice that if d = d  = 1, then v is a scalar and u is a scalar NC analytic function, hence ¯ −1 = (1 − uv) ¯ −1 (v − u). Fv (u) = (v − u)(1 − uv) Now fix v ∈ D and consider the map D → C, u → Fv (u). This map is a linear fractional map that maps the unit disc to the unit disc, maps the unit circle to the unit circle, and maps v to 0. The geometric interpretation of the map in NC variables in (1.9) is similar. Suppose we fix N ∈ N and V ∈ Bd  ×d (N ) with V  < 1 and consider the map U → FV (U ).

(1.10)

The first part of Lemma 1.6 tells us that the map defined in (1.10) maps the unit ball of d  × dtuples of N ×N matrices to the unit ball of d  ×d-tuples of N ×N matrices carrying the boundary to the boundary. The third part of Lemma 1.6 tells us that FV (V ) = 0; that is, the map given in (1.10) takes V to 0. Lemma 1.6. Suppose that N ∈ N and V ∈ Bd  ×d (N ) with V  < 1. (1) U → FV (U ) maps the Bd  ×d (N ) into itself with boundary to the boundary. (2) If U ∈ Bd  ×d (N ), then FV (FV (U )) = U . (3) FV (V ) = 0 and FV (0) = V . Proof. See Section 5.

2

1.4.2. Classification of NC ball maps General NC ball maps – those where f (0) is not necessarily 0 – are described using the linear fractional transformation F . / ∂Bd  ×d . Then Theorem 1.7. Let f : Bg  ×g → Bd  ×d be an NC ball map with f (0) ∈   f (x) = Ff (0) ϕ(x) ,

(1.11)

where   ϕ(x) = Ff (0) f (x) = V 



x 0

 0 U∗ ϕ(x) ˜

(1.12)



for some unitaries U : Cd → Cd and V : Cd → Cd and an NC analytic contraction-valued map ϕ˜ with ϕ(0) ˜ < 1. Conversely, every NC analytic f satisfying (1.11) and (1.12) for unitaries U, V and ϕ˜ as / ∂Bd  ×d . above, is an NC ball map f : Bg  ×g → Bd  ×d with f (0) ∈ Proof. Define ϕ(x) := Ff (0) (f (x)). Then ϕ(0) = 0. By Lemma 1.5, ϕ(x) is an NC analytic map. Hence it is an NC ball map sending 0 to 0 and is thus classified by Theorem 1.3. Moreover, Eq. (1.11) is implied by Lemma 1.6(2). The converse easily follows from Lemmas 1.5 and 1.6. 2 The results of Sections 1.3 and 1.4 are treated in Part I of this paper.

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1.5. More generality In this subsection we extend the main results presented so far in two directions. The first concerns LMIs. Our interest will be in properties of the set of all solutions to a given LMI. In Section 1.5.2 we will define what we mean by an LMI, then show that the set of solutions to a “monic” LMI equals a general type of matrix ball we call a pencil ball. Ultimately we would like to study maps from pencil balls to pencil balls and this paper is a beginning which handles the special case where the domain pencil ball is the ordinary NC ball Bg  ×g (see Corollary 1.10). Eventually we hope to understand which NC analytic change of variables takes one LMI to another. Work is in progress on such problems. In the next generalization we do not have applications in mind, but do something that is mathematically natural. A basic notion in several complex variables is the Shilov or distinguished boundary. A natural problem is to classify NC analytic functions mapping the ball to the ball and carrying the distinguished boundary to the boundary. Classification of linear maps of this type proves to be an interesting challenge tackled in Sections 7 and 9. For NC analytic maps we introduce the semi-distinguished boundary (a set larger than the distinguished boundary) and study the NC analytic functions mapping the semi-distinguished boundary to the boundary. All of this we only do for balls of vectors, rather than balls of matrices, that is for g = 1. 1.5.1. Linear pencils Let L(x) := A11 x11 + · · · + Ag  g xg  g

(1.13)

denote an NC analytic truly linear pencil in x. If the matrices Aij that are used to define it are  in Cd ×d , then L(x) is called a d  × d linear pencil. As an example, for g  = 2 and g = 1, 

 1 2 , A11 = 3 4



 0 1 A21 = , −1 0

the linear pencil is  L(x) =

x11 3x11 − x21

 2x11 + x21 . 4x11

1.5.2. Linear matrix inequalities and (pencil) balls Let L˜ be a d × d monic symmetric linear pencil. The positivity domain of L˜ is defined to be    ˜ 0 . DL˜ := X ∈ Mg  ×g  L(X) ˜ In other words, it is the set of all solutions to the LMI L(X)  0. We wish to analyze this solution set and we can using results on balls which we have already obtained. Now we describe DL˜ as a type of ball [11]. To do this write L˜ as L˜ = I + L + L∗ where L is a d × d NC analytic truly linear pencil, then to L(x) we associate the (pencil) ball

J.W. Helton et al. / Journal of Functional Analysis 257 (2009) 47–87

BL :=

55

∞     X ∈ Mg  ×g (n)  Idn − L(X)∗ L(X)  0 n=1

=

∞      X ∈ Mg  ×g (n)  L(X)  1 .

(1.14)

n=1

Observe that Bg  ×g = BL for L(x) =



Eij xij

i,j

with Eij being the elementary g  × g matrix with 1 located at position (i, j ). Lemma 1.8. For X ∈ Mg  ×g , 

 0 X ∈ DL˜ 0 0

iff X ∈ BL .

(1.15)

 0 X ∈ ∂DL˜ 0 0

iff X ∈ ∂BL .

(1.16)

Furthermore, 

Proof. By definition, L˜



0 X 0 0



 =

I L(X)∗

 L(X) . I

2

1.5.3. Pencil ball maps Now we turn to Bg  ×g → BL maps. As a generalization of NC ball map, given a linear pencil L, an NC analytic mapping f : Bg  ×g → BL will be called a pencil ball map provided L(f (X)) = 1, whenever X = 1 and f (X) is defined. Lemma 1.8 tells us that understanding pencil ball maps is equivalent to understanding maps on the sets of solutions to certain types of LMIs. Theorem 1.9. Let L be a d  × d NC analytic truly linear pencil and f : Bg  ×g → BL a pencil ball   map with f (0) = 0. Write h := L ◦ f . Then there exist unitaries U : Cd → Cd and V : Cd → Cd such that   x 0 (1.17) h(x) = V U ∗, ˜ 0 h(x) where h˜ is an NC analytic contraction-valued map. Proof. Follows easily by applying Theorem 1.3 to h.

2

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Corollary 1.10. Let L be a d  × d NC analytic truly linear pencil and f : Bg  ×g → BL a pencil ball map with L ◦ f (0) < 1. Then   L ◦ f (x) = FL◦f (0) ϕ(x) ,

(1.18)

where ϕ(x) = FL◦f (0) (L◦f (x)) is an NC ball map Bg  ×g → Bd  ×d taking 0 to 0 and is therefore completely described by Theorem 1.3. Proof. Apply Theorem 1.7 to L ◦ f (x).

2

1.5.4. Semi-distinguished pencil ball maps Many of our proofs with little extra effort work for a class of functions more general than pencil ball maps. These involve the notion of distinguished boundary which we now define. The Shilov boundary or distinguished boundary of Bg  ×g (N ) is the smallest closed subset  of Bg  ×g (N ) with the following property: For f : Bg  ×g (N ) → CK analytic and continuous to the boundary ∂Bg  ×g (N ), for any X ∈ Bg  ×g (N ) we have     f (X)  maxf (U ). U ∈

(1.19)

In other words, the maximum of f over Bg  ×g (N ) occurs in the distinguished boundary. We refer the reader to [15, p. 145] or [8, Ch. 4] for more details. It is a theorem [1, p. 77] that the distinguished boundary of Bg (N ) is 

  X ∈ Bg (N )  X ∗ X = I .

Accordingly, we let ∂dist Bg denote the disjoint union of these distinguished boundaries and call this the distinguished boundary of Bg . A further discussion of distinguished boundaries for Bg  ×g is in Section 6.3. An NC analytic function f : Bg → BL satisfying f (0) ∈ / ∂BL and f (∂dist Bg ) ⊆ ∂BL

(1.20)

is called a distinguished pencil ball map. Here, (1.20) means that for every isometry X for which limδ1 f (δX) exists, this limit lies in ∂BL . A natural open question is: classify distinguished pencil ball maps. Our proof of Theorem 8.1 does something like this but a little weaker. A key distinction between the semi-distinguished maps and the case treated earlier in Theorems 1.3 and 1.9 occurs with linear distinguished ball maps. These we find much harder to classify than linear NC ball maps, which we leave as an interesting open question. Definition 1.11. The semi-distinguished boundary of Bg  is defined to be 1/2

∂dist Bg  :=

∞   n=1

  1  X ∈ Bg  (n)  X ∗ X is a projection of dimension  n . 2

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An NC analytic function f : Bg  → BL satisfying f (0) ∈ / ∂BL and  1/2  f ∂dist Bg  ⊆ ∂BL

(1.21) 1/2

is called a semi-distinguished pencil ball map. Here, (1.21) means that for every X ∈ ∂dist Bg  for which limδ1 f (δX) exists, this limit lies in ∂BL . The study of semi-distinguished pencil ball maps is the subject of Part II of this article. For semi-distinguished pencil ball maps we get a weak version of the pencil ball map classification Theorem 1.9 – see Theorem 8.1. Part I. Binding 2. Models for NC contractions Let S denote the (g  -tuple of) shift(s) on noncommutative Fock space Fg  . The Hilbert space Fg  is the Hilbert space with orthonormal basis consisting of words x in g  NC variables x = (x1 , . . . , xg  ). Then Sj w = xj w for a word w ∈ x and Sj extends by linearity and continuity to Fg  . The key properties we need about S are: Sj∗ S = δj I

for j,  = 1, . . . , g  , 

I−

g 

Sj Sj∗ = P0 ,

(2.1)

j =1

where P0 is the (rank one) projection onto the span of the empty word. A column contraction is a g  -tuple of square matrices (operators), ⎡

⎤ X1 . X = ⎣ .. ⎦ Xg   such that I − X ∗ X = I − Xj∗ Xj  0. If X acts on finite dimensional space, then X is a column contraction if and only if X ∈ Bg  . In general, X is a column contraction if and only if X ∗ is a row contraction. Row contractions (and so column contractions too) are well studied – e.g. by Popescu and also Arveson. A  strict column contraction is a column contraction X for which there is an ε > 0 such that I − Xj∗ Xj  ε. If X is acting on a finite dimensional space, this last  condition is equivalent to I − Xj∗ Xj  0, i.e., X ∈ int Bg  . Column contractions are modeled by S ∗ , which is the content of Lemma 2.1 below and a major motivation for these definitions. We do not use this property of the Sj until proving Theorem 6.2. Lemma 2.1. (See [7,21].) If X is a strict column contraction acting on a Hilbert space H, then there is a Hilbert space K and an isometry V : H → K ⊗ Fg  such that V X = (I ⊗ S ∗ )V ; i.e., for each j , V Xj = (I ⊗ Sj∗ )V and in particular, for each word w ∈ x, V w(X) = (I ⊗ w(S ∗ ))V . Here I is the identity on K. Further, if X ∈ Bg  (N ) (so is a tuple of matrices), then the dimension of K can be assumed to be at most N .

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A natural generalization of the g  -tuple of shifts on Fock space to Mg  ×g and its (sequence of) ball(s) is g  ,g

X = [ Sj∗ ⊗ S ]j,=1 . (A word of caution: we have abused notation by using Sj to denote shifts on both Fg  and Fg .) The operator X should be compared to the reconstruction operator in [27]. Though we do not know if X serves as a universal model for Bg  ×g in the same way that S does for Bg  , it does serve as a type of boundary for NC analytic functions. The statement of the results requires approximating X by matrices. The operator (not matrix) X acts upon Fg  ×g – the g  ,g

Hilbert space with orthonormal basis consisting of words in g  g NC variables x = (xj, )j,=1 . Given a natural number n, let Fg (n) denote the span of words of length at most n in Fg , and set Fg  ×g (n) = Fg  (n) ⊗ Fg (n). Let Xn denote the compression of X to the (semi-invariant finite dimensional) subspace Fg  ×g (n). Lemma 2.2. Let Pn denote the projection onto the complement of the span of ∅ in Fg  (n) (and also in Fg (n)) and let Qn denote the projection onto the complement of the span of {w | w is a word of length n} in Fg  (n) (and also in Fg (n)). Then: X∗n Xn = Ig ⊗ Pn ⊗ Qn , Xn X∗n = Ig  ⊗ Qn ⊗ Pn .

(2.2)

Remark 2.3. In view of the definition of Bg  ×g , it is natural to think of an NC analytic function h on Bg  ×g as a function of the g  g variables xj, , 1  j  g  and 1    g. In turn, a monomial m in (xj, ) can be viewed as a homogeneous monomial u ⊗ v, where u and v are monomials of the same length (same as the length of m) and u and v monomials in NC variables yj (1  j  g  ) and z (1    g) respectively. In this way,    au⊗v u ⊗ v = h(α) . h= α |u|=|v|=α

α

For instance, the monomial x23 x41 is identified with y2 y4 ⊗ z3 z1 . We want to evaluate NC analytic functions Bg  ×g → Md  ×d on Xn , which is a norm one matrix thereby causing power series convergence difficulties. However, evaluating NC analytic functions on nilpotent tuples X ∈ Bg  ×g behaves especially well. Here a tuple X is called nilpotent of order β if w(X) = 0 for every word w of length  β. Lemma 2.4. If f : Bg  ×g → Md×d  is NC analytic and X ∈ Bg  ×g is nilpotent of order β, then f (X) is defined and moreover,  f (α) (X). f (X) = α<β

In particular, if f is an NC ball map, f (0) = 0, and Y ∈ ∂Bg  ×g is nilpotent of order two, then f (Y ) = f (1) (Y ).

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59

Proof. Let X ∈ Bg  ×g be given and let r denote the series radius for f . For z ∈ D with |z| < r the power series expansion for f (zX) converges. The nilpotent hypothesis gives f (zX) =



f (α) (X)zα .

αβ

Since f (zX) is analytic for |z| < 1 and is equal to the polynomial on the right-hand side above for |z| < r, equality holds for all z. If Y ∈ ∂Bg  ×g and Y is nilpotent of order two, the argument above shows f (zY ) =



f (α) (Y )zα .

α1

Moreover, the assumption f (0) = 0 implies f (0) = 0. Choosing z = 1 gives f (Y ) = f (1) (Y ).

2

Lemma 2.5. 

(a) Suppose p is an NC polynomial of degree N with Cd ×d coefficients in g  g variables and p(0) = 0. (1) If 0  I − X∗n Xn − p(Xn )∗ p(Xn ) for each n  N , then p = 0. (2) If 0  I − Xn X∗n − p(Xn )p(Xn )∗ for each n  N , then p = 0. (b) Suppose h : Bg  ×g → Md  ×d is NC analytic. If h(Xn ) = 0 for each n, then h = 0. Proof. (a) Write p=

m 

p (α)

α=0

as in Remark 2.3. In particular, p (α) =



au⊗v u ⊗ v,

|u|=|v|=α 

and au⊗v ∈ Cd ×d . By hypothesis a∅ = 0, so that p (0) = 0. Now suppose pk = 0 for k < n. Let w be a word of length n and γ ∈ Cg be given. From Lemma 2.2, we have   0 = I − X∗n Xn γ ⊗ w ⊗ ∅.

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Hence, 0 = p(Xn )γ ⊗ w ⊗ ∅ = p (n) (Xn )γ ⊗ w ⊗ ∅ =



pw⊗v γ ⊗ ∅ ⊗ v.

|v|=n

Thus pw⊗v = 0 and it follows that p (n) = 0. (b) This proof is similar. Here is a brief outline. First note that 0 = h(0). Let r denote the series radius for h. Fix N . For |z| < r and for any n  N , by Lemma 2.4 we have (since Xn is nilpotent of order n  N ) h(Xn ) =

N 

h(α) (Xn ).

α=1

 (α) of degree N , it follows from (a) that p = 0. If we now let p denote the polynomial N α=1 h Since this is true for all N , we see h = 0. 2 3. NC isometries This section has two parts. The first shows that the linear part of an NC ball map is an NC ball map, i.e., it is what is commonly known as a complete isometry. The second subsection classifies these linear NC ball maps. Recall that an NC analytic function f : Bg  ×g → Bd  ×d is an NC ball map provided it is NC analytic and contraction-valued in the interior of Bg  ×g and for X ∈ Bg  ×g (N) with X = 1, f (eit X) = 1 for almost every t ∈ R. 3.1. Pencil ball maps have isometric derivatives 



A linear mapping ψ : Cg ×g → Cd ×d is completely determined by its action on the matrix  units Ej, ∈ Cg ×g with a 1 in the (j, ) position and 0 elsewhere. The mapping ψ then naturally  extends to a mapping, still denoted ψ , on Cn×n ⊗ Cg ×g by the formula     ψ [ Xj, ]j, = Xj, ⊗ ψ(Ej, ) ∈ Cn×n ⊗ Cd ×d .

(3.1)

For notational simplicity, the formula above is written ψ(X). The mapping ψ is completely iso metric if ψ(X) = X for each X ∈ Cn×n ⊗ Cg ×g and each n, and is completely contractive if ψ(X)  X for all X. Proposition 3.1. Suppose f : Bg  ×g → Bd  ×d is an NC analytic map with f (0) = 0. If f is an NC ball map, then f (1) , the linear part of f , is a complete isometry. Proof. We start by observing that, in view of Lemma 2.4,  f for every X ∈ Bg  ×g .

(1)

0 0

X 0



 =f

0 X 0 0

 (3.2)

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61

If f is an NC ball map, then for X ∈ ∂Bg  ×g     0 X   = f   0 0

  0 1 = X =   0

X 0

   

(3.3)

by the binding property. Now by (3.2),           0 X  0 X   = f (1) f = 0     0  0 0 0 0

   f (1) (X)   = f (1) (X).  0

From (3.3) and (3.4) we obtain f (1) (X) = 1 for all X with X = 1.

(3.4)

2

Remark 3.2. This remark does not contribute to the proofs, rather it is for the sake of reconciling the definitions of complete isometries and contractions given here with what is typically found in the literature (cf. [19]).   Often a completely contractive (resp. isometric) mapping ψ : Cg ×g → Cd ×d is defined as  ×g n×n  g follows. Given n, let (C ) denote the n × n matrices with entries from Cg ×g and define  ×g n×n  ×d n×n g d 1n ⊗ ψ : (C ) → (C ) by    n 1n ⊗ ψ [ Yα,β ]nα,β=1 = ψ(Yα,β ) α,β=1 . In this definition, the block matrix Y = [ Yα,β ]nα,β=1 is written as Y=



Eα,β ⊗ Yα,β ,

where Eα,β ∈ Cn×n are the n × n matrix units. Evaluating ψ on Y becomes 1n ⊗ ψ(Y ) =



Eα,β ⊗ ψ(Yα,β ).



By using the matrix units basis Ej, of Cg ×g , Y can be rewritten as Y=



Xj, ⊗ Ej, ,

for some Xj, . Evaluating 1n ⊗ ψ on Y expressed as above gives Eq. (3.1). Passing between these two expressions for Y is known as the canonical shuffle in [19]. Letting Aj, = ψ(Ej, ), Eq. (3.1) becomes ψ(X) =



Xj, ⊗ Aj, .



3.2. Completely isometric maps on Cg ×g 

The following theorem which classifies completely isometric maps on Cg ×g is the main result of this section. It appears as Corollary 3.4 in [5] (see also [4]). For the readers convenience, we provide an elementary self-contained proof.

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Theorem 3.3. A linear mapping ψ : Cg ×g → Cd ×d is completely isometric if and only if there   exist unitaries U : Cd → Cd , V : Cd → Cd and a completely contractive (linear) mapping    ϕ : Cg ×g → C(d −g )×(d−g) such that  ψ(Y ) = V

Y 0

 0 U ∗. ϕ(Y )





Throughout this subsection let ψ : Cg ×g → Cd ×d denote a completely isometric mapping.  It is convenient to make use of the matrix units in Cg ×g . Let {ej } and {ej } denote the standard 

basis for Cg and Cg respectively. Let Aj, = ψ(ej e∗ ) for 1  j  g  and 1    g be as in Remark 3.2. We have represented ψ in terms of the matrix   g  ×g g  ,g A = [ Aj, ]j,=1 ∈ Cd ×d .

This matrix has the formal block transpose given by A∗ = [ A,j ]j, . Lemma 3.4. If ψ is completely contractive, then A∗ is a contraction. g  ,g Proof. Choose X = j,=1 ej e∗ ⊗ e (ej )∗ . Direct computation reveals that X ∗ X = Ig  g and thus the block matrix X is a contraction. Hence ψ(X) = A∗ is also a contraction.

2

Remark 3.5. (1) That the converse of Lemma 3.4 is not true in general can be seen by considering the mapping ψ : C2×2 → C defined by ψ(ej e∗ ) = δj . In this case, A∗ = I2 , but ψ(E11 + E22 ) = 2, so that ψ is not even contraction-valued. (2) For g = 1 the converse does hold. We leave this as an exercise for the interested reader. 



Proposition 3.6. A completely contractive mapping ψ : Cg ×g → Cd ×d is a complete isometry if and only if there is a set {f1 , . . . , fg } ⊆ Cd of unit vectors satisfying Aα,s fu , Aβ,t fv  =



1 0

if (α, s, t) = (β, u, v), otherwise.

Here 1  u, v  g, 1  s, t  g, and 1  α, β  g  . The following lemma is an important ingredient in the proof.

(3.5)

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63

Lemma 3.7. Under the hypotheses of Proposition 3.6, the set {f1 , . . . , fg } is orthonormal. Moreover, hα = Aα,j fj ∈ Cd



(1  α  g  )

(3.6)

is independent of j . Proof. Let fj be a set of unit vectors satisfying Eq. (3.5). Notice first that, for fixed j , the set {Aα,j fj | 1  α  g  } is an orthonormal set. Let Sj denote the span of this set. Given j,  and α, Aα,j fj =



cβ Aβ, f + ζ

for some ζ orthogonal to S (and where the dependence of the coefficients cβ on α, j,  has been suppressed). Taking the inner product with Aγ , f it follows that cβ = 1 if β = α and cβ = 0 otherwise; i.e., Aα,j fj = Aα, f + ζ. On the other hand both Aα,j fj and Aα, f are unit vectors and thus ζ = 0. Hence, Aα,j fj is independent of j and hα = Aα,j fj is unambiguously defined. Since Aα,j is a contraction (as it is, by definition, ϕ(Eα,j )) and since fj  = 1 and Aα,j fj  = 1, it follows that fj = A∗j,α hα , and is thus independent of α. Using this last claim, consider, for j = ,   2 2   A∗j,α + eit A∗,α hα  = 2 + 2 Re eit fj , f . It follows that fj , f  = 0. Here we have used   ϕ ej eα∗ ⊗ eα ej∗ + e−it e eα∗ ⊗ eα e∗ = Aα,j + e−it Aα, is a contraction, [ 1 eit ]2 = 2, and hα  = 1.

2 

Proof of Proposition 3.6. Suppose such f ’s exist. Let X ∈ Cg ×g ⊗ Cn×n with X = 1 be given. Thus X is a contraction and there is a unit vector x = xj ⊗ ej such that Xx = 1. In particular,  ∗ Xα,s xs = 1. xt∗ Xα,t

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Thus, 

ψ(X)∗ ψ(X)



xu ⊗ f u ,



u

   ∗ ∗   ∗ fv Aβ,t Aα,s fu xv∗ Xβ,t xv ⊗ f v = Xα,s xu

v

=



∗ xt∗ Xα,t Xα,s xs = 1.

Of course we also must be careful to check, in view of the orthonormality of {f1 , . . . , fg } of Lemma 3.7, 

xu ⊗ f u ,



u

  xv ⊗ f v = xu∗ xu = 1.

v

u

Thus, if X = 1, then ψ(X)  1. Since ψ assumed to be a contraction, the proof that ψ is completely isometric follows. turn to the converse. Suppose ψ is completely isometric. Fix α and choose X =  Let us now ∗ ⊗ e ⊗ (e )∗ . Then, e (e ) α  α   XX ∗ = g  eα eα∗ ⊗ eα eα∗ . Thus, ϕ(X) =



A1, ⊗ e1 (e )∗ has norm at most α = [ Aα,1

has norm at most



g  . Equivalently,

· · · Aα,g  ]

 g  . Suppose now that ⎡

⎤ h1 . h = ⎣ .. ⎦ hg  and α h2 = g  . Then, using the fact that each Aα,s is a contraction,  2         Aα,s hs , Aα,t ht  Aα,s hs  =  Aα,s hs , Aα,t ht   g = 

s,t



 s,t

s,t

Aα,s hs Aα,t ht  



hs ht  =



hs 

2

 g  h2 = g  .

s,t

The Cauchy–Schwartz inequality was used in two of the inequalities. Because equality prevails in the end, we must have equality in the inequalities. Therefore, hs 2 = g1 for each s and moreover, Aα,s hs , Aα,t ht  = for each s.

1 g

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Choose X =



 ∗ j, ej (e )

65

⊗ ej (e )∗ and note X2 = gg  . Then ϕ(X) =



 ∗ Aj, ej e

has norm squared exactly gg  . In particular, there is a unit vector ⎡

⎤ f1 f = ⎣ ... ⎦ fg  such that ϕ(X)f 2 = gg  . Hence gg  =



α f 2 .

α

From the paragraph above α 2  g  and thus for each α we must have α f 2 = g  . Again in view of the preceding paragraph, it follows that fs 2 = g1 for each s and moreover Aα,s fs , Aα,t ft  =

1 g

(3.7)

for every α, s, t. Fix α. Applying the matrix A∗ of Lemma 3.4 to the vector f1 ⊗ eα produces the vector ⎡A

α,1 f1

⎤

⎢ Aα,2 f1 ⎥ ⎢ ⎥. .. ⎣ ⎦ . Aα,g  f1 Since the first entry has norm g1 and the whole vector itself has norm at most g1 , it follows that Aα,s f1 = 0 whenever s = 1. Applying the same argument to the other indices u shows Aα,s fu = 0 for s = u. For the final ingredient, fix α = β and let  ∗  ∗ Y = eα e1 ⊗ e1∗ + eβ e2 ⊗ e2∗ . Since Y ∗ Y = e1 e1∗ ⊗ e1 e1∗ + e2 e2∗ ⊗ e2 e2∗ , Y is a contraction. Therefore, ϕ(Y ) = Aα,1 ⊗ e1∗ + Aβ,2 ⊗ e2∗

(3.8)

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is also a contraction. Let F (t) = f1 ⊗ e1 + eit f2 ⊗ e2 . With these notations, ϕ(Y )F (t) = Aα,1 f1 + Aβ,2 f2 , which gives the second equality in !   1 2= 2 + e−it Aα,1 f1 , Aβ,2 f2  + eit Aβ,2 f2 , Aα,1 f1  dt 2π !   1 ϕ(Y )F (t)2 dt  2. = 2π The inequality is a consequence of the hypothesis ϕ(Y )  1 and F (t)2 = 2. It follows that ϕ(Y )F (t) = 1 for every t and thus Aα,1 f1 , Aβ,2 f2  = 0 whenever α = β. Repeating the argument with other indices shows Aα,s fs , Aβ,t ft  = 0 if α = β. (Here s = t is ok so long as α = β.) Combining Eqs. (3.7), (3.8), and (3.9) gives the desired (3.5).

(3.9) 2

3.2.1. Characterization of complete isometries In this subsection, Theorem 3.3 is deduced from Proposition 3.6. We begin with a lemma which follows readily from Lemma 2.5. 



Lemma 3.8. Suppose the linear map Σ : Cg ×g → Cd ×d has the form   x σ1 (x) . Σ(x) = σ2 (x) σ3 (x) If Σ is a completely contractive, then σ1 = 0 and σ2 = 0. Proof. For a given n we have 0  I − Σ(Xn )∗ Σ(Xn )  I − X∗n Xn − σ2 (Xn )∗ σ2 (Xn ) = ∗

 ∗ . ∗

Thus the upper left-hand corner in the block matrix above is positive semidefinite and Lemma 2.5 implies σ2 = 0. Reversing the order of the products shows σ1 = 0. 2 Proof of Theorem 3.3. If ψ has the given form, then ψ is evidently completely isometric. Conversely, suppose ψ is completely isometric. Let fj be a set of unit vectors satisfying Eq. (3.5). By Lemma 3.7, the set {f1 , . . . , fg } is orthonormal and moreover, hα = Aα,j fj is independent of j and {h1 , . . . , hg  } is also an orthonormal set.

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67

Let F = [ f1

· · · fg ],

H = [ h1 

· · · hg  ]. 

The mappings F, H are isometries Cg → Cd and Cg → Cd respectively. Further, for given β, u, h∗β



xα,s Aα,s fu =



xα,s h∗β Aα,s fu = xα,s h∗α Aα,s fs = xα,s .

α,s

It follows that H ∗ ϕ(x)F = x. This proves the first part of this direction of the theorem. The isometries H and F extend to unitaries V and U respectively which produces the representation   x σ1 U ∗, ϕ(x) = V σ2 σ3 x σ  where the block matrix Σ = σ2 σ31 is completely contractive since the same is true for ϕ. Now Lemma 3.8 completes the proof. 2 4. Proof of Theorem 1.3 In this section we prove Theorem 1.3. Accordingly, suppose h : Bg  ×g → Bd  ×d is an NC ball map and h(0) = 0. From Lemma 3.1, h(1) , the linear part of h, is a complete isometry. By Theorem 3.3, there exist unitaries U and V and a completely contractive mapping h˜ (1) such that  h (x) = V (1)

x 0

 0 U ∗. h˜ (1) (x)

(4.1)

We claim that V h(x)U ∗ is of the desired form (1.8). For the sake of convenience we replace h(x) by V ∗ h(x)U . For X ∈ Bg  ×g (N ) consider D → Md  ×d (N), z → h(zX), which is analytic (in z). This is a function of one complex variable, so the classical Schwarz lemma applies. Hence for all 0  δ < 1 and 0  θ  2π we have  ∗   0  δ 2 I − h δeiθ X h δeiθ X . If δ is in the series radius, we may write ∞  iθ   iθ   iθ     (1) (∞) h δe X = h δe X + h δe X = h(α) δeiθ X . α=1

We integrate (4.2) for such δ to obtain

(4.2)

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1 0 2π

!2π  2  ∗   δ I − h δeiθ X h δeiθ X dθ 0

1 = δ I − δ h (X) h (X) − 2π 2

2 (1)

∗ (1)

!2π

  ∗  h(∞) δeiθ X h(∞) δeiθ X dθ

0

= δ 2 I − δ 2 h(1) (X)∗ h(1) (X) −

∞ 

δ 2α h(α) (X)∗ h(α) (X),

(4.3)

α=2

where the last equality uses the homogeneity (of order α) of h(α) .   Fix an α  2 and write δ α−1 h(α) = bb13 bb42 for NC analytic polynomials bj . Then by Eqs. (4.1) and (4.3) and because the bj are polynomials,       ∗ b1 (X) b2 (X) b1 (X)∗ b3 (X)∗ 0 I 0 X X − 0 − b2 (X)∗ b4 (X)∗ 0 h˜ (1) (X)∗ h˜ (1) (X) b3 (X) b4 (X) 0 I   I − X∗ X 0 = 0 I − h˜ (1) (X)∗ h˜ (1) (X)   b (X)∗ b1 (X) + b3 (X)∗ b3 (X) b1 (X)∗ b2 (X) + b3 (X)∗ b4 (X) . (4.4) − 1 b2 (X)∗ b1 (X) + b4 (X)∗ b3 (X) b2 (X)∗ b2 (X) + b4 (X)∗ b4 (X) 

It follows that I − X∗n Xn − bj (Xn )∗ bj (Xn )  0 for j = 1, 3 and all n. Lemma 2.5 thus implies b1 = 0 and b3 = 0. We now multiply in the other order (consider say Xn X∗n instead of X∗n Xn ) to conclude that b2 = 0 (also b1 = 0, but that we already knew). This shows h has the desired form and completes the proof. 5. Linear fractional transformation of a ball It is well known that the bianalytic maps on the unit disk D are exactly the linear fractional maps. These act transitively on the unit disk. That is, if w, z ∈ D, then there is a linear fractional map F which maps w to z. It is standard in classical several complex variables that this generalizes to special domains in Cn [8]. In this section we give basic properties of linear fractional maps on Bd  ×d . Given a d  × d matrix v with v < 1, define Fv : Bd  ×d → Bd  ×d by 1/2   −1  1/2 Fv (u) := v − Id  − vv ∗ Id − v ∗ v u Id − v ∗ u .

(5.1)

Lemma 5.1. Suppose D is an open NC domain containing 0. If u : D → Bd  ×d is NC analytic, then Fv (u(x)) is an NC analytic function (in x) on D.

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Proof. Since v is a matrix with v < 1, the expressions (Id  − vv ∗ )1/2 and (Id − v ∗ v)1/2 are constant NC analytic functions. As sums and products of NC analytic functions are NC analytic, it suffices to show that ζ := (Id − v ∗ u)−1 is NC analytic. Note that v ∗ u is NC analytic on D. Thus ζ being the composition of the NC analytic function (1 − z)−1 on D and the NC analytic function v ∗ u on D is NC analytic as well. 2 For the convenience of the reader, we now give the basic, and known (see for instance [25]) properties of F in a lemma generalizing Lemma 1.6. For this we define Uk to be the set of all U ∈ Bd  ×d (N) which are isometric on a space of dimension at least N k. For example, Ud denotes the isometries in Bd  ×d (N ). Lemma 5.2. Suppose that N ∈ N and V ∈ Bd  ×d (N ) with V  < 1. (1) U → FV (U ) maps the unit ball Bd  ×d (N ) into itself with boundary to the boundary. Furthermore, for each k  d, Uk maps onto Uk . (2) If U ∈ Bd  ×d (N ), then FV (FV (U )) = U . (3) FV (V ) = 0 and FV (0) = V . Proof. The proof is motivated by linear system theory but an understanding of system theory is not needed to read the proof. Let y ∈ CN d be given. Define  i=

i1 i2



 =

1

(I − V ∗ V )− 2 (I − V ∗ U )y −Uy





∈ CN d ⊕ CN d .

Let M denote the matrix  −V ∗ . (I − V V ∗ )1/2



(I − V ∗ V )1/2 M := V

(5.2)

Straightforward computation shows M is unitary; i.e., M ∗ M = I = MM ∗ . Let  o=

o1 o2



 = Mi =



y 1

(I − V V ∗ )− 2 (V − U )y

1



∈ CN d ⊕ CN d .

1

The relation V (I − V V ∗ )− 2 = (I − V ∗ V )− 2 V was used in computing Mi. Since M is unitary, i1 2 + i2 2 = o1 2 + o2 2 .

(5.3)

On the other hand, computations give FV (U )i1 = o2 . Combining the last two equations gives   2 2 i1 2 − FV (U )i1  = o1 2 − i2 2 = y 2  − Uy2  0.

(5.4)

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Since the mapping y → i1 = (I − V ∗ V )− 2 (I − V ∗ U )y is onto, the matrix FV (U ) is a contraction and the first part of item (1) of the lemma is proved. To prove the second part of item (1), notice that from Eq. (5.4) and the fact that both FV (U ) and U are contractions, the dimension of the space on which FV (U ) is isometric is the same as the dimension of the space on which U is isometric. We now turn to the proof of item (2). Define 1/2   −1  1/2 F := FV (U ) = V − I − V V ∗ I − V ∗V U I − V ∗U . First notice that  1/2  −1  1/2 I − V ∗V I − V ∗F = I − V ∗V + V ∗ I − V V ∗ U I − V ∗U    1/2 ∗  −1  1/2 I − V ∗V = I − V ∗V + I − V ∗V V U I − V ∗U  1/2   −1  1/2 I − V ∗U I − V ∗U I − V ∗V = I − V ∗V  1/2 ∗  −1  1/2 I − V ∗V + I − V ∗V V U I − V ∗U  1/2  −1  1/2 I − V ∗U I − V ∗V = I − V ∗V . So 

I − V ∗F

−1

 −1/2   −1/2 I − V ∗U I − V ∗V = 1 − V ∗V .

We use this and elementary calculations to obtain 1/2   −1  1/2 FV (F ) = V − I − V V ∗ I − V ∗V F I − V ∗F 1/2   −1/2   I − V ∗U =V − I −VV∗ F I − V ∗V 1/2   −1/2     I − V ∗U + I − V V ∗ U V I − V ∗V =V − I −VV∗   = V − V I − V ∗ U + U − V V ∗ U = U. For (3), compute 1/2   −1  1/2 FV (V ) = V − I − V V ∗ I − V ∗V V I − V ∗V 1/2   −1/2 =V − I −VV∗ V I − V ∗V  1/2  −1/2 I − V ∗V = V − V I − V ∗V = 0. 2 Part II. Clinging In this, and the sections to follow, we turn our attention to semi-distinguished ball maps introduced in Section 1.5.4. In particular, attention is restricted to the NC domains Bg  .

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6. NC functions revisited This section gives several basic facts about NC analytic functions on the ball, most of which are used in the remainder of the paper. We feel several of the main results here also are of interest in their own right. A few of the results are included purely for their own sake. 6.1. Series radius of convergence This section shows that NC power series expansions of NC analytic functions on a ball have good convergence properties. As a consequence of this convergence, bounded NC analytic functions are free analytic in the sense of Popescu [22]. Lemma 6.1. If h : Bg → Bd  ×d is an NC analytic function, with NC power series expansion h=



aw w,

w

then 

aw 2  d.

w

Moreover, if Z is a strict column contraction acting on a separable Hilbert space or if Z = I ⊗ S ∗ where S is the shift of Fock space Fg , and z ∈ D, then h(zZ) =



aw ⊗ (zZ)w

converges absolutely, h(zZ) is a contraction and z → h(zZ) is an analytic function on D. Proof. Let S denote the shifts introduced in Section 2. Let Fg (n) denote the span of the words of length at most n in the NC Fock space Fg . Let Wn : Fg (n) → Fg denote the inclusion. Thus, for any finite dimensional Hilbert space K, I ⊗ Sj (n) = I ⊗ Wn∗ (I ⊗ Sj )I ⊗ Wn is the compression of I ⊗ Sj to the (semi-invariant finite dimensional) subspace K ⊗ Fg (n). Here I is the identity on K. In view of the hypotheses (and since the Sj (n) are nilpotent of order n),    ∗  aw ⊗ w S(n) . h S(n)∗ = |w|n

Thus, for any vector γ ∈ Cd ,  2    2   ∗    ∗ ∗    a ∗ γ 2 . aw γ ⊗ w  = γ   h S(n) γ ⊗ ∅ =  w  2

|w|n

It follows that d

   a ∗ ej 2 , w

w

j

|w|n

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where {e1 , . . . , ed } is an orthonormal basis for Cd . (Note that the sums over j terms on the right∗ 2  hand are the squares of the Hilbert–Schmidt norms of the aw .) Since aw 2 = aw  side ∗ e 2 , it follows that a w j j d



aw 2 .

Consequently, if |z| < 1 and Z = (Z1 , . . . , Zg ) is a g tuple of operators on Hilbert space (potentially infinite dimensional) satisfying Zj∗ Zj  I and if |z| < 1, then h(zZ) :=



aw ⊗ (zZ)w

converges (absolutely). A favorite choice is Z = I ⊗ S ∗ . For |z| < 1, we have I ⊗ Wn Wn∗ h(zI ⊗ S ∗ )I ⊗ Wn Wn∗ converges in the SOT to h(zI ⊗ S ∗ ). On the other hand, I ⊗ Wn∗ h(zI ⊗ S ∗ )I ⊗ Wn = h(zI ⊗ S(n)∗ ) which is assumed to be a contraction. Thus, h(zI ⊗ S ∗ ) is a contraction. For a general strict column contraction X, represent X as V X = (I ⊗ S ∗ )V by Lemma 2.1. For |z| < 1, it follows that h(I ⊗ zS ∗ )V = V h(zX) and hence h(zX)  1. 2 6.2. The NC Schwarz lemma The classical Schwarz lemma from complex variables states the following: if f : D → D is analytic and f (0) = 0, then f (z)  z for z ∈ D. There are several ways to extend this to NC analytic functions, for example Popescu [22, Theorem 2.4] gives one and other results of this type can be found in [28]. In this subsection we give two extensions of our own. Theorem 6.2. Suppose f : Bg  → Bd  ×d is an NC analytic function on Bg  . If f (0) = 0 and f (X)  1 for each X ∈ int Bg  , then X ∗ X − f (X)∗ f (X)  0,

(6.1)

for X ∈ int Bg  . Proof. The proof relies on the model for column contractions and the convergence result for bounded NC analytic functions f of Lemma 6.1 which allows us to evaluate bounded NC analytic functions on operators, not just matrices. Since f maps into Bd  ×d , if |z| < 1, then  ∗   I − f zS ∗ f zS ∗  0,

(6.2)

by Lemma 6.1. Thus, 

 ∗   Sj Sj∗ + P0 − f zS ∗ f zS ∗  0.

(6.3)

j

(Here P0 is the projection onto the span of the empty word.) From f (0) = 0, we obtain f (S ∗ )P0 = 0. Hence (6.3) transforms into

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   ∗   (I − P0 ) Sj Sj∗ + P0 − f zS ∗ f zS ∗ (I − P0 ) j

+ P0



Sj Sj∗

  ∗ ∗  ∗  P0 + P0 − f zS f zS

j

   ∗ ∗  ∗  ∗ (I − P0 ) + P0  0. = (I − P0 ) Sj Sj + P0 − f zS f zS

(6.4)

j

As    ∗ ∗  ∗  ∗ (I − P0 ) Sj Sj + P0 − f zS f zS (I − P0 ) j

   ∗ ∗  ∗  ∗ (I − P0 ) Sj Sj − f zS f zS = (I − P0 ) j

and P0



    ∗    ∗   Sj Sj∗ − f zS ∗ f zS ∗ = 0 = Sj Sj∗ − f zS ∗ f zS ∗ P0 ,

j

j

(6.4) is equivalent to 

 ∗   Sj Sj∗ − f zS ∗ f zS ∗  0,

(6.5)

j

for |z| < 1. Replacing S by I ⊗ S in the argument above yields  ∗    Sj Sj∗ − f zI ⊗ S ∗ f zI ⊗ S ∗  0. I⊗

(6.6)

j

Given X ∈ Bg  with X < 1, we can write V X = (I ⊗ S ∗ )V , where I is the identity on a finite dimensional Hilbert space, by Lemma 2.1. Moreover, by Lemma 6.1, for |z| < 1,   Vf (zX) = f zI ⊗ S ∗ V . Multiply (6.6) by V ∗ on the left and V on the right to obtain       ∗ ∗ ∗ ∗ ∗ V V = X ∗ X − f (zX)∗ f (zX)  0, I ⊗ Sj Sj − f zI ⊗ S f zI ⊗ S j

for |z| < 1. Since X < 1, letting z  1 completes the proof.

2

Remark 6.3. Popescu [22, Theorem 2.4] formulates and proves a Schwarz lemma for free analytic functions, which in our context implies that if f is a contraction-valued NC analytic function  ∗  I for all α. (This with f (0) = 0, then f (X)  X for X < 1 and further, |w|=α aw aw inequality remains true even with operator coefficients aw .)

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A classical complex variables statement equivalent to Schwarz’s lemma is the following: if f : D → D is analytic and f (0) = 0, then h(z) = f (z) z is also analytic and h : D → D. We give a noncommutative analog of this result, which does not appear to be an immediate consequence of Theorem 6.2. Theorem 6.4. Suppose that H = [ H1 on Bg  . If for each X ∈ int Bg  ,

· · · Hg  ] is a row of d  × d NC analytic functions

      H (X)X  =  Hj (X)Xj     1,

(6.7)

j

i.e.,  ∗ I − H (X)X H (X)X  0,

(6.8)

I − H (X)H (X)∗  0.

(6.9)

then for each X ∈ int Bg 

Equivalently, H (X)  1. Proof. This proof depends upon both Lemmas 2.1 and 6.1. Let G(x) = H (x)x. The hypotheses imply G : Bg → Md×d  is contraction-valued. Hence Lemma 6.1 applies. Denote the power series expansions for Hj by Hj =



(α)

hj .

α

It follows that the power series expansion (by homogeneous terms) for G is then G=

 α

(α)

hj xj .

j

Hence, also by Lemma 6.1, for each j the power series expansion for Hj converges for any strict column contraction Z (even for operators on an infinite dimensional Hilbert space) and for such Z, G(Z) =



Hj (Z)Zj .

j

In particular, for |z| < 1 and Z = zI ⊗ S ∗ (where S is as in Lemma 2.1 and I is the identity on a finite dimensional Hilbert space),      G zS ∗ = Hj zI ⊗ S ∗ Sj∗ . j

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Because G(zI ⊗ S ∗ )  1,   0  I − G zI ⊗ S ∗ G(zI ⊗ S)∗      ∗ =I − Hj zI ⊗ S ∗ I ⊗ Sj∗ I ⊗ S H zI ⊗ S ∗ j

=I −





   ∗ Hj zI ⊗ S ∗ Hj zI ⊗ S ∗ .

(6.10)

j

Let X ∈ int Bg  be a strict column contraction acting on a finite dimensional space. Express X = V ∗ (I ⊗ S ∗ )V according to Lemma 2.1, where I is the identity on a finite dimensional Hilbert space. For every NC analytic polynomial f and |z| < 1, f (zX) = V ∗ f (zI ⊗ S ∗ )V . Hence the same holds true for NC analytic functions and in particular,   Hj zI ⊗ S ∗ V = V Hj (zX). Thus, applying V on the right and V ∗ on the left of Eq. (6.10) gives  0I − Hj (zX)Hj (zX)∗ . j

Letting z  1 concludes the proof.

2

6.3. The distinguished boundary for Bg  ×g Fix N . The distinguished (Shilov) boundary of the algebra A(Bg  ×g (N )), the functions which are analytic in int Bg  ×g (N ) and continuous on Bg  ×g (N ) is the smallest closed subset  of Bg  ×g (N ) so that each element of A(Bg  ×g (N )) takes its maximum on . That a smallest, as opposed simply minimal, such set exists is a standard fact in complex analysis and the theory of uniform algebras; see [15, p. 145] or [8, Ch. 4] for more details. While not needed in the sequel, the following known result explains the distinguished terminology in the definitions of distinguished isometry and semi-distinguished pencil ball map. Proposition 6.5. The distinguished boundary of A(Bg  ×g (N )) is {X ∈ Bg  ×g (N ) | X ∗ X = I }. That the distinguished boundary of A(Bg  ×g (N )) must be contained in {X ∈ Bg  ×g (N ) | X ∗ X = I } follows readily from Lemma 5.2; see Proposition 6.6. For the fact that no smaller set can serve as a distinguished boundary, we refer the reader to [1, p. 77]. 

Proposition 6.6. Fix N ∈ N. If f : Bg  ×g (N ) → Cd ×d is continuous and analytic in int Bg  ×g (N ), then for any X ∈ Bg  ×g (N ) we have     f (X)  max f (U ) (6.11) U ∈Uk

for any 0 < k  min{g  , g}. Thus if f (X) = 0 for all X ∈ Bg  ×g (N ) such that X ∗ X = I (if g   g) or XX ∗ = I (if g  < g), then f = 0. For example, if g   g, then the set of isometries Ug contains the distinguished boundary of Bg  ×g (N ).

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Proof. First suppose f : Bg  ×g (N ) → C (so that d = 1 = d  ). Pick any U ∈ Uk . By the maximum principle, the function h(z) = f (zU ) takes its maximum value on |z| = 1. Now we use linear fractional automorphisms of the ball to prove that such an inequality holds for any X in the interior of Bg  ×g (N ). Select F as in Lemma 5.2 which maps 0 to X. Then h(Z) := f (F (Z)) is analytic and maps 0 to f (X). The previous paragraph applies to give          f (X) = h(0)  maxh(zU ) = maxf F (zU ) . |z|=1

|z|=1

By Lemma 5.2(1), F (zU ) ∈ Uk for |z| = 1; so we have proved that the maximum of f occurs on Uk . To prove the statement for matrix-valued f , simply note that given unit vectors γ ∈ Cd and  η ∈ Cd , the function F (X) = η∗ f (X)γ takes it maximum on Uk . It follows that     F (X)  max f (U ). U ∈Uk

Since γ and η are arbitrary, the result follows.

2

Remark 6.7. This proposition has more content for larger k and in particular k = min{g, g  } is optimal. 6.4. Matrix Linksnullstellensatz For scalar NC analytic polynomials there is an elegant Linksnullstellensatz whose proof is due to Bergman, cf. [9]. Now we generalize it to matrices with entries which are NC analytic polynomials. Theorem 6.8. Given an m × d matrix P over Cx and an n × d matrix Q over Cx, suppose that P (X)v = 0 implies Q(X)v = 0 for every matrix g-tuple X and vector v. Then for some G ∈ Cxn×m we obtain Q = GP . Proof. The rows of a matrix A will be denoted by Aj = [ aj 1 aj 2 · · · aj d ]. In particular, Pj is a 1 × d matrix over Cx. Let Vd = Cx1×d denote the left Cx-module of 1 × d matrices of polynomials. Note Pj ∈ Vd . Let Id be the Cx-submodule of Vd generated by the Pj , i.e., Id =



 "  rj Pj  rj ∈ Cx .

Id is the smallest subspace of Vd containing the Pj and invariant with respect to Mj = left multiplication by xj (for each j ). Note that Mj determines a well-defined linear mapping Yj on the quotient: Yj : Vd /Id → Vd /Id . Let Wk denote the image of polynomials of degree at most k in the quotient Vd /Id . These spaces are finite dimensional and Wk−1 ⊆ Wk . So Wk−1 is complemented in Wk .

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Choose N > max degree of all polynomials in P and Q. Define Xj = Yj : WN −1 → WN and extend Xj to a linear mapping WN → WN in any way (on a complementary subspace). Let vj denote the element # of WN determined by the row with the polynomial 1 in the j -entry and 0 elsewhere. Define v = vj ∈ WNd . For a polynomial q, q(X)vj = [ 0 · · · 0 q 0 · · · 0 ] (j th spot). Hence Qj (X)v = Qj . A similar statement is true for Pj ; i.e., Pj (X)v = Pj ∈ Id and so Pj (X)v = 0. So Qj (X)v = Qj is 0 too which means Qj ∈ Id . Thus there exists Gsj such that Qj = Hence Q = GP , as desired.



Gj s Ps .

2

7. The linear part of semi-distinguished ball maps Having established preliminary results, we turn our attention to semi-distinguished ball maps, introduced in Section 1.5.4. First we show that semi-distinguished ball maps have very distinctive linear parts. And then we set about to give properties of these linear maps.  A linear map L : Cg → Cd ×d is a distinguished isometry if it maps the distinguished boundary of Bg to the boundary of Bd  ×d ; i.e., if for each X ∈ Bg with X ∗ X = I we have that L(X) = 1. In this case a (nonzero) vector γ such that L(X)γ  = γ  is called a clinging vector and this property clinging. Proposition 7.1. If f is a semi-distinguished ball map, then f (1) , the linear part of f , is a distinguished isometry. Proof. The proof is the same as that of Proposition 3.1.

2

7.1. Properties of distinguished isometries The remainder of this section is devoted to giving properties of distinguished isometries. 

Proposition 7.2. Let L : Cg → Cd ×d be a linear map. (1) L is a distinguished isometry if and only if   L (X) := X1∗ X1 + · · · + Xg∗ Xg ⊗ Id − L∗ (X)L(X)  0

(7.1)

and clings (i.e., L (X) is always positive semidefinite and never positive definite). (2) If L is completely isometric, then it is a distinguished isometry. The converse is not true. Proof. For the implication (⇒) in (1), given any Xi , choose a W satisfying W ∗ W = X1∗ X1 + · · · + Xg∗ Xg . Note that it suffices to show (7.1) on a dense subset of Bg  . Thus we may assume that W is invertible. Then (X1 W −1 )∗ X1 W −1 + · · · + (Xg W −1 )∗ Xg W −1 = I , so by assumption, I − L∗ (XW −1 )L(XW −1 )  0 and it binds. Since L is truly linear, we multiply this inequality with W ∗ on the left and with W on the right: W ∗ W − L∗ (X)L(X)  0 and it binds. The converse (⇐) is obvious.

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First part of (2) is trivial. To finish the proof it suffices to exhibit an example of a distinguished isometry which is not a complete isometry. Consider L(x, y) = Ax + By with ⎤ 0 0 ⎥ ⎦, 0 √

⎡

1 √0 ⎢ 0 22 A=⎣ 0 0 0 0  1 0  0 0 For X = 0 0 and Y = 1 0 ,

⎡ ⎢ B =⎣

0 √ 2 2

2 2

0 0

⎤ 0 0 ⎥ ⎦. 0 √

0 0 1 0

2 2

$     X  √ 3       Y  = 2 > 2 = L(X, Y ) . This shows that L is not a complete isometry. It remains to be seen that L satisfies (7.1). We compute ⎡

1 ∗ 2y y ⎣ L (x, y) = − 12 x ∗ y

0

− 12 y ∗ x 1 ∗ 2x x 0

⎤ 0 ⎦. 0 1 ∗ 2 (x − y) (x − y)

The top left 2 × 2 block of L (x, y) can be factored as  ∗ −∗  1 ∗   −1 −y x 1 −x y 2x x 0 1 0 0 0 1

 1 . 0

This immediately implies that for invertible X, L (X, Y ) is always positive semidefinite and never positive definite. For noninvertible X the same holds true by a standard density argument. 2 

Remark 7.3. By way of contrast, every contractive L : Cg → Cd ×d is completely contractive. For related results see Section 9. 7.1.1. The Gram representation A powerful tool used is a matrix representation of a quadratic NC polynomial. A key property of this representation is that matrix positivity of the quadratic NC polynomial is equivalent to the positive semidefiniteness of the representing matrix. The following lemma is needed to establish this. Lemma 7.4. For large enough n the set  g    Xw  X ∈ Cn×n , w ∈ Cn

(7.2)

is all Cng . Proof. Given w, x1 , . . . , xg ∈ Cn with xj = 0, choose Xj ∈ Cn×n such that Xj w = xj . For w∗ instance Xj = xj w 2 2 will do. Note Lemma 7.4 is true even with a fixed w = 0 and parametrizing over all X.

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Proposition 7.5. Let p=



xi∗ Bij xj

1i,j g 

be a homogeneous quadratic NC polynomial with Bij ∈ Cd ×d . Then there is a unique matrix  G ∈ (Cd ×d )g×g with p = x ∗ Gx. Moreover, p(X) = iff G  0.

 i,j

(7.3)

Bij ⊗ Xi∗ Xj is positive semidefinite for all N ∈ N and all X ∈ (CN ×N )g

Proof. In d  × d block form, G = [ Bij ]i,j . If G  0, then G = H ∗ H for some matrix H . Hence p = (H x)∗ (H x) is a sum of hermitian squares, so p(X)  0 for all N ∈ N and X ∈ (CN ×N )g . The converse follows from Lemma 7.4. 2 7.1.2. Orthotropicity In this subsection we establish a basic property of distinguished isometries L (that is, of those L for which L is positive semidefinite and clinging), which we call orthotropicity.  A d  × d linear pencil L = A1 x1 + · · · + Ag xg : Cg → Cd ×d is called orthotropic if for every X ∈ Cg and w ∈ Cd satisfying L(X)w = w, the vector L(X)w is orthogonal to the image of L(X ⊥ ). Proposition 7.6. Every distinguished isometry is orthotropic. To continue our analysis of distinguished isometries we write L in a special form. We multiply L with a unitary V on the left and a unitary U ∗ on the right. Thus without loss of generality, A1 is the block matrix   1 0 (7.4) 0 (A1 )22 and Aj for j  2 equals 

0 (Aj )21

 (Aj )12 . (Aj )22

(7.5)

g Proof of Proposition 7.6. Suppose L = i=1 Ai xi is a distinguished isometry and without loss of generality write L in the special form described above. Clearly, orthotropicity of L is equivalent to (Aj )12 = 0 for j  2. In order to prove this we set all variables except for X1 , Xj to 0. For convenience we use X, Y (resp. x, y) instead of X1 , Xj (resp. x1 , xj ) and A, B instead of A1 , Aj . Thus  L(x, y) =

x B21 y

 B12 (Id−1 ⊗ y) . A22 (Id−1 ⊗ x) + B22 (Id−1 ⊗ y)

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∗ ∗ ∗ A straightforward computation shows we can  x ∗  x   x represent L (x, y) = x x +y y −L(x, y) L(x, y) as y G y (cf. Proposition 7.5), where y stands for

⎡ x   x ⎢0 =⎣ y y 0

0

⎤

Id−1 ⊗ x ⎥ ⎦ 0 Id−1 ⊗ y

and ⎡

0 ⎢ 0 G=⎢ ⎣ 0 ∗ −B12

0 I − A∗22 A22 ∗ A −B21 22 ∗ A −B22 22

0

−A∗22 B21 ∗ B 1 − B21 21 ∗ −B22 B21

⎤ −B12 ⎥ −A∗22 B22 ⎥. ∗ ⎦ −B21 B22 ∗ ∗ I − B12 B12 − B22 B22

If L (X, Y ) is positive semidefinite for all X, Y , then by Proposition 7.5, G is positive semidefinite. In particular, B12 = 0. (Note if L (X, Y ) is only positive semidefinite for scalars X, Y , then B12 need not be 0.) Alternative proof of B12 = 0. By density, we may assume Y is invertible. L (X, Y ) multiplied on  m m12   −1  the right by Y 0 and on the left by the transpose of the same matrix yields M2 := m11 21 m22 0 I for ∗ m11 = 1 − B21 B21 , ∗ ∗ m12 = −B21 A22 (Id−1 ⊗ X) − B21 B22 (Id−1 ⊗ Y ) + Y −∗ X ∗ B12 (Id−1 ⊗ Y ),   ∗  ∗    B21 + Id−1 ⊗ Y ∗ B12 XY −1 , m21 = m∗12 = − Id−1 ⊗ X ∗ A∗22 B21 − Id−1 ⊗ Y ∗ B22   ∗   ∗ m22 = Id−1 ⊗ X X + Y ∗ Y − Id−1 ⊗ Y ∗ B12 B12 (Id−1 ⊗ Y )       ∗ − Id−1 ⊗ X ∗ A∗22 + Id−1 ⊗ Y ∗ B22 A22 (Id−1 ⊗ X) + B22 (Id−1 ⊗ Y ) . (7.6)

Consider m12 and note that Y −∗ X ∗ B12 (Id−1 ⊗ Y ) = Y −∗ X ∗ Y B12 . Suppose B12 = 0. Then it is easy to construct X = X(ε), Y = Y (ε) sending this term to ∞ as ε → 0 while keeping all the remaining terms bounded. This contradiction yields B12 = 0. 2 8. Characterization of semi-distinguished ball maps The following theorem summarizes what we know about semi-distinguished pencil ball maps. Both the hypotheses and the conclusions are weaker than those of Theorem 1.9. The relationship between Theorem 1.9 and Theorem 8.1 is made precise by Corollary 8.2 of this section. Theorem 8.1. Let L be a d  × d NC analytic truly linear pencil and f : Bg  → BL a semidistinguished pencil ball map with f (0) = 0. Clearly, h := L ◦ f maps Bg  → Bd  ×d . Write h h(1) is the linear homogeneous component in the NC power series as h = h(1) + h(∞) , where  (∞) (α) d  ×d )g  and a = ∞ expansion of h and h α=2 h . Then there is a unique contraction M ∈ (C  unique nontrivial subspace S ⊆ Cdg such that:

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(1) h(1) (x) = Mx, M|S is an isometry and MΠS ⊥ is a strict contraction. (2) Each h(α) (x) for α  2 is of the form Pα Π S ⊥ x for a matrix Pα of NC ∞polynomials. (∞) (X)v := (3) For the formal NC power series P (∞) := ∞ P , P α=2 α α=2 Pα (X)v converges for v ∈ S ⊥ ⊗ CN and X ∈ (CN ×N )g in an NC ε-neighborhood of 0. Also, h(∞) (x) = P (∞) (x)ΠS ⊥ x. (4) (M ⊗ IN + P (∞) (X))ΠS ⊥ ⊗CN   1 for X ∈ (CN ×N )g with X < 1 and (M ⊗ IN ) × (S ⊗ CN ) is orthogonal to (M ⊗ IN )(S ⊥ ⊗ CN ), to Pα (X)(S ⊥ ) for all α  2 and to P (∞) (X)(S ⊥ ). Proof. Let h(1) (x) := x ∗ x − h(1) (x)∗ h(1) (x) = x ∗ Gx be as in Proposition 7.5, where G  0. Write h(1) (x) = Mx and note that G = 1 − M ∗ M. Let S := ker G = ker(I − M ∗ M) = range(I − M ∗ M)⊥ . By the clinging property, S = {0}. By definition, M|S is an isometry. Conversely, if v satisfies Mv = v, then Mv, Mv = v, v and hence v, (I − M ∗ M)v = 0. Since M is a contraction, I − M ∗ M is positive semidefinite. Thus (I − M ∗ M)v = 0, that is, v ∈ S. This proves (1) and also the uniqueness of M and S. For (2) fix N  1 and let X ∈ Bg  (N ) such that X = 1 be given. By Eq. (6.1) of Schwarz’s lemma (Theorem 6.2) applied to h(zX), |z| < 1 for all 0  δ < 1 and 0  θ  2π we have  ∗   0  δ 2 X ∗ X − h δeiθ X h δeiθ X .

(8.1)

If series radius, we may write h(δeiθ X) = h(1) (δeiθ X) + h(∞) (δeiθ X) = ∞δ is (α)in the iθ α=1 h (δe X). We integrate (8.1) to obtain 1 0 2π

!2π  2 ∗  ∗   δ X X − h δeiθ X h δeiθ X dθ 0

1 = δ X X − δ h (X) h (X) − 2π 2

∗

2 (1)

∗ (1)

!2π

  ∗  h(∞) δeiθ X h(∞) δeiθ X dθ

0

= δ 2 X ∗ X − δ 2 h(1) (X)∗ h(1) (X) −

∞ 

h(α) (δX)∗ h(α) (δX).

(8.2)

α=2

By Proposition 7.2, h(1) (X)  0 with clinging. Thus by (8.2), for every w satisfying  ∗  X X − h(1) (X)∗ h(1) (X) w = 0

(8.3)

we have h(α) (δX)w =√0 for α  2 and δ in the series radius. In particular, by Proposition 7.5, (8.3) is equivalent to Gxw = 0 and this implies that h(α) (X)w = 0 for α  2 and every X in the series radius. By a scaling argument (h(α) is homogeneous), the same holds true for every X and w. Hence the matrix NC Nullstellensatz that there is √ Theorem 6.8 applies √ and implies √ ˜α with P˜α (x) Gx = h(α) (x). Since G = G(ΠS + ΠS ⊥ ) = a√matrix of NC polynomials P √ G ΠS ⊥ , we set Pα = P˜α G. Then h(α) (x) = Pα (x)ΠS ⊥ x. (3) The second part is clear and for the first statement we refer the reader to [14].

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(4) Let v ∈ S and w ∈ S ⊥ . Then % & Mv, Mw = M ∗ Mv, w = v, w = 0

(8.4)

since (1 − M ∗ M)v = 0. This shows that M(S ⊥ ) is orthogonal to M(S). For the strengthening with tensor products, let si ∈ S, ti ∈ S ⊥ , vi , uj ∈ CN . Then         (M ⊗ IN ) si ⊗ vi , (M ⊗ IN ) tj ⊗ uj = (Msi ⊗ vi ), (Mtj ⊗ uj ) i

j

i

j

 = Msi , Mtj vi , uj  = 0. i,j

˜ Let h(x) = h(x)x, where 

˜ h(x) =M +

Pα (x)ΠS ⊥ .

α2

˜ h(X) ˜ By Theorem 6.4 (applied with H = h),  1 for all X with X < 1. ˜ Rewrite h(x) as    ˜ h(x) = MΠS + M + Pα (x) ΠS ⊥ .

(8.5)

α2

Both summands have norm  1 for X with X < 1. In particular,        M ⊗ IN + Pα (X) ΠS ⊥ ⊗CN     1, α2

as desired. ˜ ˜ Clearly, h(x)| ⊗ CN ) is orthogonal to S = M|S is an isometry and thus h(X)(S ˜h(X)(S ⊥ ⊗ CN ) = (M ⊗ IN + P (∞) (X))(S ⊥ ⊗ CN ). Since (M ⊗ IN )(S ⊗ CN ) is orthogonal to (M ⊗ IN )(S ⊥ ⊗ CN ), this implies (M ⊗ IN )(S ⊥ ⊗ CN ) ⊥ P (∞) (X)(S ⊥ ⊗ CN ). Suppose w ∈ (M ⊗ IN )(S ⊥ ⊗ CN ). Then w ∗ P (∞) (tX)(S ⊥ ⊗ CN ) = {0} for small enough t. Then 0 = w ∗ P (∞) (tX) = w ∗



t α Pα (X) =

α2

implies (M ⊗ IN )(S ⊥ ⊗ CN ) ⊥ Pα (X)(S ⊥ ⊗ CN ).



  t α w ∗ Pα (X)

α2

2

Let us note in passing that under the conditions of the previous theorem, (8.3) implies h(∞) (X)w = 0 for all X ∈ Bg . Indeed, let us consider the analytic function z → h(∞) (zX)w on D. Clearly, (8.3) holds for X replaced by δX due to homogeneity. If δ is in the series radius,

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then by the NC power series expansion and the lemma, h(∞) (δX)w =

∞ 

h(α) (δX)w = 0.

α=2

Thus by analytic continuation, h(∞) (X)w = 0. Next we give a corollary which makes the relationship between Theorems 8.1 and 1.9 clearer. Corollary 8.2. Keep the assumptions and notation from Theorem 8.1. If, in addition, f is a pencil ball map, then M is a complete isometry. Conversely, h = L ◦ f satisfying (1), (2), (3), (4) for a complete isometry M is an NC ball map Bg  → Bd  ×d sending 0 to 0. Proof. Suppose f is a pencil ball map. Then h(1) is a (linear) NC ball map by Proposition 3.1. Hence h(1) = Mx with M a complete isometry (see Theorem 3.3). ˜ ˜ For the converse, suppose h satisfies (1)–(4). By (1) and (3), h(x) = h(x)x, where h(x) is given by    ˜h(x) = MΠS + M + Pα (x) ΠS ⊥ . α2

˜ ˜ is a (1) and (4) imply that h(X) is for X ∈ Bg  an orthogonal sum of two contractions, thus h(X) ˜ is a contraction. contraction for X ∈ Bg  , i.e., X  1. Hence h(X) = h(X)X For the binding property of h we use that M is a complete isometry. Let e denote the disj tinguished vector associated with M, that is, A∗j Ai e = δi e if M = [ A1 · · · Ag  ] (cf. Proposition 3.6). By (1), h(1) (X) binds at e ⊗ w, where w is a binding vector for I − X ∗ X, i.e., X ∗ Xw = w. This concludes the proof since h(X)(e ⊗ w) = h(1) (X)(e ⊗ w) by (2) and (3). 2 9. Further analysis of distinguished isometries We have successfully classified complete isometries, see Theorem 3.3. Distinguished isometries are more challenging and a few sample results are provided below. Theorem 9.1. Suppose L is an orthotropic linear pencil in 2 variables. If L (X1 , X2 )  0 for all X1 , X2 ∈ Cn×n , and clings for all scalar X1 , X2 ∈ C, then L (X1 , X2 ) clings for all X1 , X2 ∈ Cn×n . Remark 9.2. We conjecture based on inconclusive computer experiments that Theorem 9.1 is false in 3 variables. 9.1. Equations which reformulate the clinging property Throughout this subsection L will denote an orthotropic d  × d linear pencil in g variables. We assume it clings for X ∈ Cg . Let L (x) = x ∗ x − L(x)∗ L(x) = x ∗ Gx

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be the Gram representation as in Proposition 7.5. Given G ∈ (Cd×d )g×g we call the linear subspace of its kernel spanned by all the vectors of the form ⎡

⎤ α1 v ⎣ ... ⎦ ∈ ker G αg v the scalar binding kernel N0 . (Since L clings for X ∈ Cg , for every α1 , . . . , αg ∈ C there exists ' α1 v ( .. d a 0 = v ∈ C with . ∈ ker G.) αg v

Fix a basis ⎫ ⎧ ⎡α v ⎤ i,1 i ⎬  ⎨  ηi := ⎣ ... ⎦  i = 1, . . . , t + m ⊆ Cgd ⎭ ⎩ αi,g vi for the scalar binding kernel N0 of G. We assume that {v1 , . . . , vt } is a maximal linearly independent set and that vt+j =

t 

γj i v i

i=1

for j = 1, . . . , m. Let X1 , . . . , Xg ∈ Cn×n . We assume that X1 is invertible and define Zi := X1−1 Xi . (Matrix) binding at X is equivalent to the existence (for all Zi ) of a nontrivial solution to L (In , Z2 , . . . , Zg )v = 0. This is implied by the existence of ri ∈ Cn for which there is a nonzero v ∈ Cdn such that ⎡

⎤ (Id ⊗ In )v  ⎢ (Id ⊗ Z2 )v ⎥ t+m ⎢ ⎥= ηi ⊗ ri . . ⎣ ⎦ .. (Id ⊗ Zg )v

(9.1)

i=1

In particular, v=

t+m  i=1

=

t  i=1

vi ⊗ ri = /

t 

1

vi ⊗ ri +

t 

j =t+1 i=1

i=1

vi ⊗ ri +

t+m 

t+m  j =t+1

23

Γi (r)

0

γj i rj . 4

γj i vi ⊗ rj

J.W. Helton et al. / Journal of Functional Analysis 257 (2009) 47–87

Similarly, (Id ⊗ Zk )v =

t

i=1 vi

t 

85

⊗ Γi (Zk r). Using this in (9.1) yields

vi ⊗ Γi (Zk r) =

i=1

t 

  vi ⊗ Γi r diag(αk ) ,

i=1

where r = [ r1 · · · rt+m ] and diag(αk ) is the diagonal matrix with αi,k as its (i, i) entry. Linear independence of the v1 , . . . , vt gives Γi (Zk r − r diag(αk )) = 0 for all k = 2, . . . , g and i = 1, . . . , t. Thus for all these i, k: t+m 

(Zk − αi,k )ri +

γj i (Zk − αj,k )rj = 0.

(9.2)

j =t+1

Hence if all the Zk − αi,k are invertible, ri = −

t+m 

  bij diag(αk ), Zk rj ,

(9.3)

j =t+1

where   bij diag(αk ), Z := γj i (Z − αi,k )−1 (Z − αj,k ). Equations derived so far reformulate then clinging property and we say how precisely in the following lemma. Lemma 9.3. Consider the following conditions: (i) For each Z2 , . . . , Zg the system of Eqs. (9.2) has a solution r1 , . . . , rt+m ; (ii) L clings. Then (i) ⇒ (ii) and if N0 = ker G, then (ii) ⇒ (i). Proof. Follows from the computations given above.

2

9.2. The general case and the proof of Theorem 9.1 Now we give a theorem more general than Theorem 9.1 that implies Theorem 9.1. Theorem 9.4. Suppose t (g − 2) < m. If L (X)  0 for all X ∈ (Cn×n )g and clings for all scalar X ∈ Cg , then L (X) clings for all X ∈ (Cn×n )g . Proof. We assume all bij (diag(αk ), Zk ) exist, i.e., all Zk − αi,k are invertible. This causes no loss of generality: the set of all matrix g-tuples that make L cling is closed and our condition implies clinging on a dense subset. Eq. (9.3) gives ri = ri (k) as a function of k. By Lemma 9.3 we need to show that for every choice of Zi the system (9.2) has a solution, i.e., ri (2) = ri (3) = · · · = ri (g) for all i = 1, . . . , t.

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This yields tn(g − 2) homogeneous equations in mn unknowns. Thus if m > t (g − 2) this system will always have a nontrivial solution. 2 Proof of Theorem 9.1. Fix a basis 

  cαi vi   i = 1, . . . , t + m βi vi

for the scalar binding kernel N0 , where {v1 , . . . , vt } is a maximal linearly independent set. In view of Theorem 9.4 it suffices to show that m > 0. Suppose m = 0 and choose α, β with βα = αβii for all i. By scalar binding, there is a nonzero  α u vector u with βii u ∈ N0 , i.e., for some λi : 

    t t  αv αu = λi ηi = λi i i . βu βi vi i=1

i=1

Hence β

t  i=1

λi αi vi = α

t 

λi βi vi

i=1

and thus by the linear independence of the vi , βλi αi = αλi βi for all i = 1, . . . , t. As at least one λj is nonzero, this implies α αj = , β βj contrary to our assumption. Thus m > 0, as desired.

2

Acknowledgments The authors thank Victor Vinnikov and Dima Kalyuzhnyi-Verbovetski˘ı for helping us with NC analytic functions. References [1] L.A. A˘ızenberg, Carleman’s Formulas in Complex Analysis: Theory and Applications, Kluwer, 1993. [2] J.A. Ball, V. Bolotnikov, Q. Fang, Schur-class multipliers on the Fock space: de Branges–Rovnyak reproducing kernel spaces and transfer-function realizations, in: Operator Theory, Structured Matrices, and Dilations, in: Theta Ser. Adv. Math., vol. 7, 2007, pp. 85–114. [3] J.A. Ball, G. Groenewald, T. Malakorn, Bounded real lemma for structured noncommutative multidimensional linear systems and robust control, Multidimens. Syst. Signal Process. 17 (2006) 119–150. [4] D.P. Blecher, D.M. Hay, Complete isometries – An illustration of non-commutative functional analysis, in: Proceedings of 4th Conference on Function Spaces, in: Contemp. Math., vol. 328, 2003, pp. 85–97. [5] D.P. Blecher, D.M. Hay, Completely isometric maps into C-star algebras, preprint http://arxiv.org/abs/math/ 0203182. [6] D.P. Blecher, C. Le Merdy, Operator Algebras and Their Modules – An Operator Space Approach, Cambridge University Press, 2004. [7] A.E. Frazho, Complements to models for noncommuting operators, J. Funct. Anal. 59 (3) (1984) 445–461.

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[8] S. Helgason, Differential Geometry Lie Groups and Symmetric Spaces, Academic Press, 1978. [9] J.W. Helton, S.A. McCullough, A Positivstellensatz for non-commutative polynomials, Trans. Amer. Math. Soc. 356 (2004) 3721–3737. [10] J.W. Helton, S.A. McCullough, M. Putinar, V. Vinnikov, Convex matrix inequalities versus linear matrix inequalities, IEEE Trans. Automat. Control, in press. [11] J.W. Helton, S.A. McCullough, V. Vinnikov, Noncommutative convexity arises from linear matrix inequalities, J. Funct. Anal. 240 (2006) 105–191. [12] D. Kalyuzhnyi-Verbovetski˘ı, Carathéodory interpolation on the non-commutative polydisk, J. Funct. Anal. 229 (2005) 241–276. [13] D. Kalyuzhnyi-Verbovetski˘ı, V. Vinnikov, Non-commutative positive kernels and their matrix evaluations, Proc. Amer. Math. Soc. 134 (2006) 805–816. [14] D. Kalyuzhnyi-Verbovetski˘ı, V. Vinnikov, Foundations of noncommutative function theory, in preparation. [15] S.G. Krantz, Function Theory of Several Complex Variables, Wiley, 1982. [16] P.S. Muhly, B. Solel, Schur class functions and automorphism of Hardy algebras, Doc. Math. 13 (2008) 365–411. [17] P.S. Muhly, B. Solel, The Poisson kernel for Hardy algebras, Complex Anal. Oper. Theory, in press. [18] E. Nelson, The distinguished boundary of the unit operator ball, Proc. Amer. Math. Soc. 12 (1961) 994–995. [19] V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2002. [20] G. Pisier, Introduction to Operator Space Theory, Cambridge University Press, 2003. [21] G. Popescu, Isometric dilations for infinite sequences of noncommuting operators, Trans. Amer. Math. Soc. 316 (1989) 523–536. [22] G. Popescu, Free holomorphic functions on the unit ball of B(H)n , J. Funct. Anal. 241 (2006) 268–333. [23] G. Popescu, Free holomorphic functions and interpolation, Math. Ann. 342 (2008) 1–30. [24] G. Popescu, Free pluriharmonic majorants and commutant lifting, J. Funct. Anal. 255 (2008) 891–939. [25] G. Popescu, Hyperbolic geometry on the unit ball of B(H )n and dilation theory, Indiana Univ. Math. J. 57 (2008) 2891–2930. [26] G. Popescu, Noncommutative transforms and free pluriharmonic functions, Adv. Math. 220 (2009) 831–893. [27] G. Popescu, Unitary invariants in multivariable operator theory, Mem. Amer. Math. Soc., in press. [28] G. Popescu, Free holomorphic automorphisms of the unit ball of B(H )n , J. Reine Angew. Math., http://arxiv.org/ abs/0810.0451, in press. [29] D.-V. Voiculescu, Free analysis questions I: Duality transform for the coalgebra of ∂X:B , Int. Math. Res. Not. 16 (2004) 793–822. [30] D.-V. Voiculescu, Free analysis questions II: The Grassmannian completion and the series expansions at the origin, preprint http://arxiv.org/abs/0806.0361. [31] M.R. Wohlers, Lumped and Distributed Passive Networks. A Generalized and Advanced Viewpoint, Academic Press, 1969.

Journal of Functional Analysis 257 (2009) 88–121 www.elsevier.com/locate/jfa

Uniform K-homology theory Ján Špakula Mathematisches Institut, Universität Münster, Einsteinstr. 62, 48149 Münster, Germany Received 17 October 2008; accepted 9 February 2009 Available online 25 February 2009 Communicated by Alain Connes

Abstract We define a uniform version of analytic K-homology theory for separable, proper metric spaces. Furthermore, we define an index map from this theory into the K-theory of uniform Roe C∗ -algebras, analogous to the coarse assembly map from analytic K-homology into the K-theory of Roe C∗ -algebras. We show that our theory has a Mayer–Vietoris sequence. We prove that for a torsion-free countable discrete group Γ , the direct limit of the uniform K-homology of the Rips complexes of Γ , limd→∞ K∗u (Pd Γ ), is isomortop phic to K∗ (Γ, ∞ Γ ), the left-hand side of the Baum–Connes conjecture with coefficients in ∞ Γ . In particular, this provides a computation of the uniform K-homology groups for some torsion-free groups. As an application of uniform K-homology, we prove a criterion for amenability in terms of vanishing of a “fundamental class”, in spirit of similar criteria in uniformly finite homology and K-theory of uniform Roe algebras. © 2009 Elsevier Inc. All rights reserved. MSC: primary 46L80 Keywords: Analytic K-homology; Coarse assembly map; Uniform Roe algebra

1. Introduction The analytic K-homology theory of a second countable locally compact Hausdorff topological space X (see e.g. [8]) can be understood as an attempt to organize the elliptic differential operators over the space X into an abelian group. The (higher) indices of these operators can be interpreted as K-theory elements over C ∗ X, the Roe C∗ -algebra [12]. The Coarse Baum–Connes E-mail address: [email protected]. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.02.008

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and coarse Novikov conjectures assert certain properties of this index (or coarse assembly) map μ : K∗ (X) → K∗ (C ∗ X), and have applications in geometry (see e.g. [12,15]). Also, the Coarse Baum–Connes conjecture can be viewed as an algorithm to compute the K-theory of Roe C∗ -algebras. In this spirit, the work presented here is setting up a framework for obtaining an algorithm to compute the K-theory groups of uniform Roe C∗ -algebras. In this paper, we define a refined version of analytic K-homology theory. We loosely follow the exposition [8] of analytic K-homology. The main idea is to quantify “how well approximable by finite rank operators” are various compact operators appearing in the definition of a Fredholm module. Our theory, compared to the classical K-homology, has some advantages (the theory becomes sensitive to some coarse properties, for instance amenability), but also some disadvantages (the K-theory of uniform Roe algebras tends to be uncountable if nonzero). The theory exhibits similarities to the uniformly finite homology theory of Block and Weinberger [3,4], which should be connected to our theory via a Chern character map. This is analogous to the Chern map from analytic K-homology into the locally finite homology groups. Using estimates from [11], we show that some elliptic operators coming from geometry give rise to uniform K-homology classes. Furthermore, we construct an index map μu from uniform K-homology into the K-theory of uniform Roe C∗ -algebras. The original example of a coarse index theorem [11] is actually carried out in this uniform context. We prove that amenability of a metric space is equivalent to non-vanishing of a “fundamental class” in the uniform K-homology of the space. Our criterion is parallel to similar criteria in the uniformly finite homology [3] and K-theory of uniform Roe algebras [6]. Our proof borrows ideas from both of these papers. In the case when the space in question is a Cayley graph of a countable torsion–free group Γ , we show that limd→∞ K∗u (Pd Γ ), the direct limit of uniform K-homologies of its Rips complexes top is naturally isomorphic to K∗ (Γ, ∞ Γ ), the left-hand side of the Baum–Connes conjecture for the group Γ with coefficients in ∞ (Γ ). This is analogous to a result of Yu [14], where he shows the equivalence of the Coarse Baum–Connes conjecture and the Baum–Connes conjecture with coefficients in ∞ (Γ, K ). This statement is true without any assumption on torsion; it is open whether the torsion-free assumption can be dropped in the uniform setting. On the other hand, since the Baum–Connes conjecture with commutative coefficients is known for a number of torsion–free groups, this result provides a computation of some uniform K-homology groups. The structure of this paper is as follows: Section 2 introduces the uniform K-homology groups, and in Section 3 we prove that certain Dirac-type differential operators give rise to uniform K-homology classes. Sections 4 and 5 are devoted to proving the Mayer–Vietoris sequence in our theory. We turn to coarse geometry and the index map in Sections 7–9. The connection between the uniform K-homology of a group Γ and the Baum–Connes conjecture with coefficients in ∞ Γ is shown in Section 10. In the final Section 11, we prove our criterion for amenability. 2. Uniform K-homology groups In this paper, the spaces are separable proper metric spaces, unless explicitly specified otherwise. Throughout the paper, X shall stand for such a space, and d will denote its metric. Finally, to avoid set-theoretic difficulties, all Hilbert spaces are assumed to be separable. Recall the definition of Fredholm modules—the representatives of cycles in the classical analytic K-homology theory.

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Definition 2.1. Let (H, φ, S) be a triple, where H is a Hilbert space, φ : C0 (X) → B(H ) is a ∗ -homomorphism and S ∈ B(H ) is an operator. We say that such a triple is a 0-Fredholm module (or even Fredholm module), if for every f ∈ C0 (X) the following hold: • (Fredholmness) (1 − S ∗ S)φ(f ) ∈ K (H ) and (1 − SS ∗ )φ(f ) ∈ K (H ), • (pseudolocality) [S, φ(f )] ∈ K (H ). Similarly, we say that a triple (H, φ, P ) as above is a 1-Fredholm module (or odd Fredholm module), if for every f ∈ C0 (X): • (P 2 − 1)φ(f ) ∈ K (H ) and (P − P ∗ )φ(f ) ∈ K (H ), • (pseudolocality) [P , φ(f )] ∈ K (H ). Remark 2.2. We can also formulate the Fredholmness condition for even Fredholm modules in another form, which is more convenient for the setting of differential operators: A triple (H, φ, T ) forms an even Fredholm module, if H is Z2 -graded, φ(f ) is of degree 0 (i.e. even) for all f ∈ C0 (X) and T ∈ B(H ) is a pseudolocal operator of degree 1 (odd), satisfying that (T 2 − 1)φ(f ) ∈ K (H ) and (T ∗ − T )φ(f ) ∈ K (H ) for all f ∈ C0 (X). We modify this concept, defining “uniform Fredholm modules”, which shall represent elements in the uniform K-homology theory. We introduce uniformity by “quantifying” the compactness of an operator in the following way: given ε > 0 we try to approximate our compact operator within ε by a finite rank operator with the smallest possible rank. In the definition of a Fredholm module, instead of just one compact operator, we really have a collection of compacts, depending on f ∈ C0 (X), and we require a uniform bound on the ranks of ε–approximants for fixed R—a “scale” in the metric of X. This consideration is sufficient to ensure uniformity on the large scale. However, we want (certain) first order differential operators to give rise to uniform K-homology classes. The approximation properties of compacts arising from the pseudolocality condition really depend not only on the support of f but also on its derivative (just consider an operator [D, f ]), and so we need to build in also some local control. Specifically, for a metric space X and R, L  0 we denote      CR (X) = f ∈ Cc (X)  diam supp(f )  R and f ∞  1 ,    CR,L (X) = f ∈ CR (X)  f is L-continuous . We say that f : X → Y is L-continuous, if there exists a nondecreasing function α: [0, ∞] → [0, ∞) with α  (0)  L1 , such that for any x, y ∈ X we have d(x, y)  α(s) ⇒ d(f (x), f (y))  s. Loosely, one could formulate the condition as “locally L-Lipschitz”. In particular, if a function is L-Lipschitz, then it is L-continuous (with α(s) = L1 s). The converse is true for instance when X is a geodesic space. Hence for practical purposes we can replace this condition with just L-Lipschitz. We use the notion of L-continuity to emphasize the local side of being Lipschitz. The reason for introducing L-continuity is the following: if X is a manifold and f ∈ CR,L (X) is differentiable at x ∈ X, then the norm of the derivative df of f at x is at most L. This observation is used in a crucial way in Section 3, when proving that Dirac-type operators produce uniform K-homology classes. If one doesn’t require the theory to include such classes, it is possible to just ignore L’s and l-’s throughout the paper.

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 Furthermore L0 CR,L (X) is dense in CR (X). This is completely analogous to saying that (once) differentiable functions are dense in the space of all continuous functions. The proof is outlined at the end of this section, in Lemma 2.18. In the following definition, we introduce the uniformity conditions. We list two versions—one without the “L-dependency” and one featuring L. Definition 2.3 (Uniform approximability). Let H be a Hilbert space, X a metric space and φ: C0 (X) → B(H ) a ∗ -homomorphism. For ε, M > 0, an operator T ∈ B(H ) is said to be (ε, M)approximable, if there is a rank-M operator k, such that T − k < ε. Let E(·) (or E(f )) stand for an expression with operators in B(H ) and terms φ(·) (or φ(f )). (For instance E(·) = T φ(·) or E(f ) = [T , φ(f )].) • For ε, M, R > 0, an expression E(·) is said to be (ε, R, M; φ)-approximable, if for each f ∈ CR (X), E(f ) is (ε, M)-approximable. • For ε, R, L, M > 0, an expression E(·) is said to be (ε, R, L, M; φ)-approximable, if for each f ∈ CR,L (X), E(f ) is (ε, M)-approximable. • An expression E(·) is uniformly approximable, if for every R  0, ε > 0 there exists M > 0, such that E(·) is (ε, R, M; φ)-approximable. Furthermore, we write E1 (·) ∼ua E2 (·), if the difference E1 (·) − E2 (·) is uniformly approximable. • An expression E(·) is l-uniformly approximable, if for every R, L  0, ε > 0 there exists M > 0, such that E(·) is (ε, R, L, M; φ)-approximable. Furthermore, we write E1 (·) ∼lua E2 (·), if the difference E1 (·) − E2 (·) is l-uniformly approximable. We introduce a special cases of uniform approximability: • We say that an operator T ∈ B(H ) is uniform, if T φ(·) and φ(·)T are uniformly approximable (i.e. T φ(f ) ∼ua 0 ∼ua φ(f )T ). We also say that T is (ε, R, M; φ)-uniform, if both operators φ(f )T , T φ(f ) are (ε, R, M; φ)-approximable. • An operator T ∈ B(H ) is said to be uniformly pseudolocal, if [T , φ(·)] is uniformly approximable (i.e. [T , φ(f )] ∼ua 0). • An operator T ∈ B(H ) is said to be l-uniformly pseudolocal, if [T , φ(·)] is l-uniformly approximable (i.e. [T , φ(f )] ∼lua 0). Remark 2.4. The property of being uniformly pseudolocal is obviously stronger than that of being l-uniformly pseudolocal. In the former, we can obtain a bound M on ranks of approximants, which is independent of L (local condition), and depends only on R (support condition) and of course on ε. Remark 2.5. The notion of an “l-uniform” operator is in fact equivalent to the notion of a uniform operator given above. More precisely, if T φ(·) and φ(·)T are l-uniformly approximable, then they are in fact just uniformly approximable, i.e. we can get a bound on M independent of L. In other words, checking uniformity of operator on “nice” function is sufficient. Indeed, for every f ∈ CR (X) we can construct a function f˜ ∈ CR+1,1 (X), such that f f˜ = f . Now given R, ε > 0, if M is the constant such that T φ(·) and φ(·)T are (ε, R + 1, 1, M; φ)-approximable, then φ(f )T = φ(f )φ(f˜)T and T φ(f ) = T φ(f˜)φ(f ) are (ε, R, M; φ)-approximable. Such an f˜ can be constructed for instance as f˜(x) = max(0, 1 − d(x, supp(f ))). One easily checks that this function is 1-Lipschitz, and so f˜ ∈ Cdiam(supp(f ))+1,1 (X).

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In the view of the previous remark, we can completely disregard the constant L appearing in the definition above, when we work with uniform operators only. This is so for instance in Sections 6–11. Definition 2.6. Let (H, φ, S) be a 0-Fredholm module. It is said to be uniform, if S is l-uniformly pseudolocal and the operators 1 − SS ∗ , 1 − S ∗ S are uniform. Let (H, φ, Q) be a 1-Fredholm module. It is said to be uniform, if Q is l-uniformly pseudolocal and the operators 1 − Q2 and Q − Q∗ are uniform. Remark 2.7. By using “uniform Fredholm module” (without 0- or 1-) in a statement we shall mean that the statement applies to both 0- and 1-uniform Fredholm modules. Remark 2.8. If we are given a Hilbert space H together with a ∗ -homomorphism φ : C0 (X) → B(H ) (i.e. an action of C0 (X) on H ), we say that (H, φ), or just H , is an X-module. When no confusion about φ can arise, we identify f ∈ C0 (X) with φ(f ) ∈ B(H ). Similarly, we omit “φ” from (ε, R, M; φ), etc. Example 2.9 (Fundamental class). Let Y be a uniformly discrete space. Let S be the unilateral shift operator on 2 N (i.e. a Fredholm operator with index 1). Denote H = 2 Y ⊗ 2 N, and set S˜ = diag(S) ∈ B(H ). Endow H with the multiplication action of C0 (Y ). More precisely, define φ : C0 (Y ) → B(H ) by φ(f )(ζ (y)) = f (y)ζ (y), for ζ : Y → 2 N, a square summable function, and y ∈ Y , f ∈ C0 (Y ). ˜ is a 0-uniform Fredholm module for Y (S˜ is actually uniIt is easy to check that (H, φ, S) formly pseudolocal). This module has pivotal role in our characterization of amenability in Section 11. The following example is concerned with the K-homology classes coming from elliptic differential operators on manifolds. Example 2.10. Let M be a complete Riemannian manifold and S a smooth complex vector bundle over M. Let D be a symmetric elliptic differential operator operating on sections of S. Let χ : R → R be a chopping function (an odd smooth function), χ(t) > 0 for t > 0, χ(t) → ±1 for t → ±∞. Denote H = L2 (M, S) and let ρ : C0 (M) → B(H ) be the multiplication action. It is proved in [8, Section 10.8], that (H, ρ, χ(D)) is a Fredholm module (whether odd or even depends on the dimension of M). Assuming that M has bounded geometry and that the operator D is “geometric” (e.g. has finite propagation speed), this Fredholm module is actually uniform. The proof is outlined in Section 3. We now proceed towards the definition of uniform K-homology groups. Definition 2.11. A collection (H, φt , St ), t ∈ [0, 1], of uniform Fredholm modules is a homotopy, if: • t → St is continuous in norm,

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• the C∗ -algebras Θ(φt ) ⊂ B(H ) generated by all the uniform operators with finite propagation1 all the same for all t ∈ [0, 1]. By an operator homotopy we mean a homotopy as above, with the restriction that φt = φ0 for all t ∈ [0, 1]. Remark 2.12. The second condition above is satisfied if for instance φt ’s fulfill • there exists R > 0, such that for f, g ∈ C0 (X) with d(supp(f ), supp(g))  R, we have φs (f )φt (g) = 0 for all s, t ∈ [0, 1], • for every s, t ∈ [0, 1] and R > 0, there are R  and M, so that every φt (f ), f ∈ CR (X), is within a rank-M operator from one of the form φs (g), g ∈ CR  (X). We now proceed as in [8, Section 8.2] in defining a K-homology theory. Given two uniform Fredholm modules, we can clearly form their direct sum, which becomes again a uniform Fredholm module. Definition 2.13 (K∗u ). We define the uniform K-homology group Kiu (X), i = 0, 1, to be an abelian group generated by the unitary equivalence classes of uniform i-Fredholm modules (H, φ, S) with the following relations: • if two uniform Fredholm modules x, y are homotopic, we declare [x] = [y], • for two uniform Fredholm modules x, y, we set [x ⊕ y] = [x] + [y]. Recall that a Fredholm module (H, φ, S) is called degenerate, if the conditions in the definition hold exactly, that is (1 − S ∗ S) = (1 − SS ∗ ) = [φ(f ), S] = 0 for all f ∈ C0 (X) for the 0-version; and S − S ∗ = S 2 − 1 = [φ(f ), S] = 0 for all f ∈ C0 (X) for the 1-version. The K∗u class of a degenerate Fredholm module is 0: the proof of the analogous result for K-homology [8, 8.2.8] carries over verbatim. The additive inverse of [(H, φ, S)] ∈ K0u (X) is [(H, φ, −S ∗ )]. Similarly, the additive inverse of [(H, φ, P )] ∈ K1u (X) is [(H, φ, −P )]. Again, the proof of these facts is just as [8, proof  tS sin tI  , t ∈ [0, π2 ], is a homotopy showing that [(H, φ, S)] + of 8.2.10]. For instance, cos sin tI − cos tS ∗    = 0 ∈ K0u (X). [(H, φ, −S ∗ )] = H ⊕ H, φ ⊕ φ, I0 0I It follows from the facts in the last two paragraphs, that every element of K∗u (X) can be represented as a class of a single uniform Fredholm module. Furthermore, [x] = [y] in K∗u (X) if and only if there exists a degenerate Fredholm module z, such that x ⊕ z and y ⊕ z are unitarily equivalent to a pair of homotopic uniform Fredholm modules. In this case, we say that x and y are stably homotopic. Therefore, we may reformulate the definition of K∗u (X) as follows: Proposition 2.14. The group Kiu (X) is canonically isomorphic to the semigroup of stable homotopy equivalence classes of uniform i-Fredholm modules. 1 Recall (see 6.4) that T ∈ B(H ) has finite propagation, if there exists R  0, such that for any f, g ∈ C (X) whose 0 supports are at least R apart, we have φt (f )T φt (g) = 0.

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The uniform K-homology is not functorial under continuous maps in general; we need two extra condition in order to obtain functoriality: one handling the large-scale and one taking care of the local phenomena. Definition 2.15. (See also 6.1.) A (not necessarily continuous) map g : X → Z between metric spaces X and Z is said to be uniformly cobounded, if for any r  0, we have    Rg (r) := sup diam g −1 B(z, r) < ∞. z∈Z

Observe that an L-continuous uniformly cobounded map g : X → Z descends to a homomorphism on the uniform K-homology groups g∗ : K∗u (X) → K∗u (Z) by the following observation: Take a uniform Fredholm module (H, φ : C0 (X) → B(H ), S) of an K∗u (X)-element. We denote by g˜ : C0 (Z) → C0 (X) the induced ∗ -homomorphism. Then there is a ∗ -homomorphism φ ◦ g˜ : C0 (Z) → B(H ). By uniform coboundedness, we obtain that if f ∈ CR (Z), then ˜ ) ∈ CRg (R),LL (X). Hence the g(f ˜ ) ∈ CRg (R) (X). By L-continuity, f ∈ CR,L (Z) implies g(f uniformity requirements transfer and (H, φ ◦ g, ˜ S) becomes a uniform Fredholm module repre˜ S)]. senting a K∗u (Z)-element. We define g∗ [(H, φ, S)] = [(H, φ ◦ g, We now prove a simple lemma analogous to a similar statement in the classical K-homology: Lemma 2.16. If (H, φ, T ) is a uniform Fredholm module and K ∈ B(H ) is uniform, then (H, φ, T ) and (H, φ, T + K) are operator homotopic. Proof. We need to show that (H, φ, T + tK), t ∈ [0, 1] are uniform Fredholm modules. Fix ε, R, L > 0 and let M be such that all the operators K, [T , φ(f )], (1 − T ∗ T )φ(f ) and (1 − T T ∗ )φ(f ) (or (1 − T 2 )φ(f ) and (T − T ∗ )φ(f ) in the 1-case) are (ε, M)–approximable for f ∈ CR,L (X). First, for f ∈ CR,L (X), we have that [T + tK, φ(f )] = [T , φ(f )] + tKφ(f ) − tφ(f )K, which is clearly (3ε, 3M)-approximable. Hence the pseudolocality requirement is satisfied. Let us now deal with the 0-case. Examine the following expression: 1 − (T + tK)(T + tK)∗ = (1 − T T ∗ ) − tKT ∗ − tT K ∗ − t 2 KK ∗ . Taking f ∈ CR,L (X) and multiplying by φ(f ) the previous formula on the right, each of the elements (1 − T T ∗ )φ(f ), tT K ∗ φ(f ), t 2 KK ∗ φ(f ) is going to be (T Kε, M)-approximable by assumption. We can rewrite the remaining term as follows tKT ∗ φ(f ) = tKφ(f )T ∗ + tK[T ∗ , φ(f )], and so it is (2T Kε, R, 2M)-approximable. Therefore, (1 − (T + tK)(T + tK)∗ )φ(f ) is (5T Kε, 5M)-approximable. It is clear that similar considerations can be applied to 1 − (T + tK)∗ (T + tK) as well. Finally, we deal with the 1-case. Let f ∈ CR,L (X). Observe that ((T + tK) − (T ∗ + tK ∗ ))φ(f ) is (2ε, 2M)-approximable. Furthermore, (1 − (T + tK)2 )φ(f ) = (1 − T 2 )φ(f ) − tT Kφ(f ) − t 2 K 2 φ(f ) − tKφ(f )T − tK[T , φ(f )] and this last expression is (5T Kε, 5M)-approximable. 2 As a first application of the previous lemma, we make the following observation: Remark 2.17. We may always assume that a K1u -element is represented by a uniform 1-Fredholm module (H, φ, Q) with Q selfadjoint. It is because if we take any Q, 12 (Q + Q∗ ) is selfadjoint and Q − 12 (Q + Q∗ ) = 12 (Q − Q∗ ) is uniform. Moreover, the procedure of replacing Q by a selfadjoint operator can be applied to whole homotopies.

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We finish the section by a lemma promised earlier. Lemma 2.18. Let X be a metric space. Given any compactly supported continuous function f : X → C and ε > 0, there exists L > 0 and an L-continuous function g : X → C, such that f − g∞ < ε. Proof. Without loss of generality we can assume that f (X) ⊂ [0, 1]. Take an integer N , such that N1 < ε, and set Un = f −1 [0, Nn ], n = 0, . . . , N . Then U0 ⊂ U1 ⊂ · · · ⊂ UN = X are closed sets. By uniform continuity, there exists δ > 0, such that d(x, y) < δ implies |f (x) − f (y)| < N1 . This implies that Nδ (Un ) ⊂ Un+1 . / Un(x)−1 Define g : X → R as follows: for x ∈ X, let n(x) be such that x ∈ Un(x) , but x ∈ 1 1 (where we set U−1 = ∅). Now set g(x) = n(x)−1 + · min(1, d(x, U )) if n(x) > 0, and n(x)−1 N N δ 1 g(x) = 0 if n(x) = 0. It is clear from the construction that f − g∞  N < ε and it is easy to verify that g is N1δ < εδ -continuous. 2 3. Dirac-type operators In this section, we outline the proof of the fact that “geometric” operators on complete Riemannian manifolds with bounded geometry give rise to uniform Fredholm modules. The hard work was already done in [11], where it is shown that such geometric operators have index defined in the algebraic K-theory of the algebra U−∞ (M) of operators given by smooth uniformly bounded kernels, the precursor of the uniform Roe algebra. Recall the setting: Let M be a complete Riemannian manifold (without boundary) and S a Clifford bundle over M. More precisely, denote by Cliff(M) the complexified bundle of Clifford algebras Cliff(Tx M) (equipped with a natural connection), and let S be a smooth complex vector bundle over M equipped with an action of Cliff(M) and a compatible connection. The bundle S is graded, if in addition it is equipped with an involution anticommuting with the Clifford action of tangent vectors (see Remark 2.2). A “geometric” operator will be a first-order differential operator D defined by the composition Γ (S) → Γ (T ∗ M ⊗ S) → Γ (T M ⊗ S) → Γ (S), where the arrows are given by the connection, metric and Clifford multiplication, respectively. In local coordinates, this operator has the form D=



ek

∂ . ∂xk

The signature and Dirac operators are of this type. The main properties of these operators is that they are elliptic, and have finite propagation (in a sense that there exists a constant C, such that supp(eitD ξ ) ⊂ NCt (supp(ξ )) for all ξ ∈ Γc (S)).2 Denote H = L2 (S) and let ρ : C0 (M) → B(H ) be the multiplication action. Let χ : R → R be a chopping function (an odd smooth function, χ(t) > 0 for t > 0, χ(t) → ±1 for t → ±∞). Then (H, ρ, χ(D)) is a Fredholm module (see [8, Sections 10.6 and 10.8]). This is true in more 2 Recall that N (Y ) denotes the δ-neighborhood of a set Y ; and Γ (S) denotes the set of smooth sections of S. δ

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general context, namely for any first-order elliptic differential operator on a complex smooth vector bundle. However to obtain uniformity, bounded geometry assumption and some analysis from [11] is required. Following [11, Section 2], we say that a Riemannian manifold M has bounded geometry, if it has positive injectivity radius and the curvature tensor is uniformly bounded, as are all its covariant derivatives. A bundle S has bounded geometry, if its curvature tensor, as well as all its covariant derivatives, are uniformly bounded. By [11, Proposition 2.4], bounded geometry can be seen by existence of nice coordinate patches, such that the Christoffel symbols comprise a bounded set in the Fréchet structure on C ∞ . For the record, let us collect all the assumptions and the conclusion into a theorem. Theorem 3.1. Let D be a geometric operator (as described above) on a Clifford bundle S over a complete Riemannian manifold with bounded geometry. For any chopping function χ , the triple (L2 (S), ρ, χ(D)) is a uniform Fredholm module. The idea of the proof (which will be made more precise afterward) is as follows: it is proved in [11, Theorem 5.5], that if ϕ ∈ C0 (R) satisfies ϕ (k) (t)  Ck (1 + |t|)m−k , then ϕ(D) extends to a continuous map between Sobolev spaces W r → W r−m for any r. Now a bounded piece of our manifold can be transferred to a torus. The Fourier coefficients of a W −k -function on a torus decay faster than s → s1k . Hence the finite rank approximants to the inclusion W r−m → W r can be constructed just by truncating the Fourier series—and knowing the rate of decay of the coefficients tells us how big rank do we need for a given ε > 0—independently on the position of our bounded piece in the manifold. Putting the facts together, ϕ(D) : W r → W r−m → W r is uniformly approximable. In order to cite [11, Theorem 5.5], we need to introduce some notation. Define (global) Sobolev spaces W k (S) as the completion of Γc (S) in the norm  2 1/2  ξ k = s2 + Ds2 + · · · + D k s  . Furthermore, if L ⊂ M, denote    ξ k,L = inf ζ k  ζ ∈ W k (S), ξ = ζ on a neighborhood of L . An operator A : W k (S) → W l (S) is called quasilocal, if there exists a function μ : R+ → R+ , such that μ(r) → 0 as r → ∞ and for each K ⊂ M and each ξ ∈ W k (S) supported within K one has Aξ l,M\Nr (K)  μ(r)ξ k . We call μ a dominating function for A. Finally, we set S m (R) to be the set of functions ϕ ∈ C ∞ (R), which satisfy inequalities of the form  (k)    ϕ (λ) < Ck 1 + |λ| m−k and define the Schwartz space S (R) =



S m (R).

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Theorem 3.2. (See [11, Theorem 5.5].) Let D be a geometric operator on a Clifford bundle S over a complete manifold M with bounded geometry. If ϕ ∈ S m (R), then ϕ(D) continuously extends to a quasilocal operator W r (S) → W r−m (S). Proof of Theorem 3.1. Fix now a function ϕ ∈ S m (R) (m  −1). We are going to show that ϕ(D) is a uniform operator.3 By the above theorem, there is a dominating function μ for ϕ(D), and ϕ(D) extends to a bounded operator L2 (S) → W −m (S). Fix now also ε > 0 and R > 0. Pick any open subset U ⊂ M with diam(U )  R. Consider now he restriction of ϕ(D) to sections supported on U (denoted L2 (S|U )). This is sufficient to obtain uniformity, since ϕ(D) is selfadjoint. Since μ(r) → 0, there is r0 > 0, such that μ(r0 ) < ε/2. Now decompose ϕ(D)|L2 (S|U ) : L2 (S|U ) → W −m (S|Nr0 (U ) ) ⊕ W −m (S|M\Nr0 (U ) ). By quasilocality, the second component has norm at most ε/2. Hence the argument is finished by proving that the restrictions of ϕ(D) to L2 (S|U ) → W −m (S|Nr0 (U ) ) → L2 (S|Nr0 (U ) ) are approximable by finite rank operators, such that the ranks depend only on ε > 0 and R  diam(U ). We can now reduce to the case of a torus with a trivial bundle. This just follows from a partition of unity argument and the existence of nice coordinate patches (from bounded geometry). Also note that for a given R, there is a uniform bound on how many of these patches are needed to cover any subset of M with diameter less than R + 2r0 . On the torus T n with the trivial bundle E = T n × Cn , we can use Fourier series. Denote by PN : L2 (E) → L2 (E) the orthogonal projection given by replacing the q -Fourier coefficient ( q ∈ Zn ) of a function by 0 if | q | > N (in other words, we truncate the Fourier series at N ). The absolute values of the Fourier coefficients of a function in W −m (E) decrease at least as PN

fast as | q |m . Consequently, the finite–rank maps W −m (E) → L2 (E) −→ L2 (E) approximate the inclusion W −m (E) → L2 (E) in norm for m  −1. Moreover, for a given ε > 0, the rank of an ε-approximant depends only on ε and m. This concludes the proof of the fact that ϕ(D) is uniform if ϕ ∈ S m (R) with m  −1. The passage from ϕ ∈ S m (R), m  −1, to ϕ ∈ C0 (R) is by the usual approximation argument (together with the fact that uniform operators from a C∗ -algebra, see 4.2). Summarizing, for ϕ ∈ C0 (R) we have that ϕ(D) is a uniform operator. Now if χ is any chopping function, then χ(D)2 − 1 = (χ 2 − 1)(D) and χ 2 − 1 ∈ C0 (R), hence the Fredholmness condition follows from the previous argument. Furthermore, the difference of two chopping functions is also in C0 (R), and so we are free to choose one particular chopping function (we choose χ(t) = √ t 2 ) to prove that χ(D) is l-uniformly pseudolocal. We apply a 1+t

useful formula from [9, Lemma 4.4]:

2 χ(D) = π

∞ 0

D dλ 1 + λ2 + D 2

(convergence in the strong topology), so that

3 Note that the notion of a uniform operator from [11] is different from ours.

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 2 ρ(f ), χ(D) = π

∞ 0

  1 1 + λ2 ρ(f ), D 1 + λ2 + D 2

  + D ρ(f ), D D

1 1 + λ2

+ D2

dλ.

Fix ε > 0 and L > 0. We have estimates 1 •  1+λD 2 +D 2   2λ , • for a smooth f ∈ CR,L (M), [ρ(f ), D] is the multiplication operator by the derivative of f , and so we have that [ρ(f ), D]  L.

Consequently, the integral in the last display converges in norm; and there exists k > 0 and λ1 , . . . , λk , such that the integral can be approximated within ε > 0 by the sum of the integrands 1 with λ = λ1 , . . . , λk . Now each of the operators 1+λD 2 +D 2 , 1+λ2 +D 2 is uniform by the previous considerations:

t 1 , 1+λ2 +t 2 1+λ2 +t 2

∈ S −1 (R). This finishes the proof.

2

We finish the section by an observation, which can be applied to obtain uniform Fredholm modules for non-geometric elliptic operators. We assume that a finitely generated discrete group Γ acts cocompactly on M (this assumption implies that M has bounded geometry), and that D commutes with this action. The vague reason for uniformity is that D “looks the same” on each translate of a fundamental domain (which is bounded), and so the approximation properties of D at any place of M are the same as those over a fixed fundamental domain. In this case, just knowing that ϕ(D) is locally compact for ϕ ∈ C0 (R) upgrades to: Claim 1. For any ϕ ∈ C0 (R), the operator ϕ(D) is uniform. Proof. For a given R > 0, we can find a bounded open set U ⊂ M, such that the collection {U γ }γ ∈Γ covers M and has Lebesgue number at least R. Construct a continuous function f : M → [0, 1], which is 1 on U and 0 outside a small neighborhood of U . Then for any function g ∈ CR (M) there is a γ ∈ Γ , such that g · f γ = g (by f γ we denote the translate of f by γ ). Then ρ(g)ϕ(D) = ρ(gf γ )ϕ(D) = ρ(g)ρ(f γ )ϕ(D γ ) = ρ(g)(ρ(f )ϕ(D))γ . Hence (ε, N)-approximability of ρ(g)ϕ(D) is not worse than the one of ρ(f )ϕ(D) (which is a compact operator, independent of g). This proves that ϕ(D) is uniform. 2 Pseudolocality can be now deduced in the same way as in the geometric case from the claim, provided that [ρ(f ), D] is bounded independently of f ∈ CR,L (M). 4. Dual algebras In the analytic K-homology, one can use the Voiculescu’s theorem and a standard normalizing procedure to express K-homology as a K-theory of a certain C∗ -algebra. In this section, we first work on a fixed X-module (H, φ) to obtain a similar isomorphism for the “partial” uniform K-homology groups (Proposition 4.3). To work around the Voiculescu’s theorem, we express the uniform K-homology as a direct limit of “partial” uniform K-homology groups (Proposition 4.9).

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Definition 4.1. Let H be a Hilbert space and let φ : C0 (X) → B(H ) be a ∗ -representation. We define Ψφ0 (X) ⊂ B(H ) to be the set of all l-uniformly pseudolocal operators in B(H ) and Ψφ−1 (X) ⊂ B(H ) to be the set of all uniform operators. Furthermore, we denote 0 (X) ⊂ B(H ⊕ H ). Dφu (X) = Ψφ⊕0

Lemma 4.2. Let H be a Hilbert space and φ : C0 (X) → B(H ) a ∗ -representation. Then Ψφ0 (X) ⊂ B(H ) is a C∗ -algebra. Likewise, Ψφ−1 (X) ⊂ Ψφ0 (X) is a C∗ -algebra. Furthermore, Ψφ−1 (X) is a closed two-sided ideal of Ψφ0 (X).

Proof. We show that Ψφ0 (X) is norm-closed. Assume that T ∈ B(H ) is approximable by l-uniformly pseudolocal operators. Take ε > 0 and R, L  0. By assumption, there is an luniformly pseudolocal operator S ∈ B(H ), such that T − S < ε/4. Let M be such that S is (ε/2, R, L, M; φ)-approximable. Hence for any f ∈ CR,L (X) there exists k ∈ B(H ) with rank(k)  M such that [φ(f ), S] − k < ε/2. Consequently, [φ(f ), T ] − k  [φ(f ), (T − S)] + [φ(f ), S] − k < ε. In other words, [φ(f ), T ] is (ε, M)-approximable. The proof that the norm-limits of uniform operators are again uniform is analogous. The identity [φ(f ), ST ] = [φ(f ), S]T + S[φ(f ), T ] implies that Ψφ0 (X) is closed under multiplication. Likewise, using the identity φ(f )ST = [φ(f ), S]T + Sφ(f )T we obtain that Ψφ−1 (X) is an ideal of Ψφ0 (X) (we’re using Remark 2.5 here). 2 For a fixed X-module (H, φ), define a group K∗u (X; φ) in a similar manner as K∗u (X), except that we consider only (unitary equivalence classes of) uniform Fredholm modules, whose Hilbert spaces and C0 (X)-actions are direct sums (finite or countably infinite) of (H ⊕ H, φ ⊕ 0). A glance at the proofs for K∗u (X) shows that K∗u (X; φ) can be characterized also as a group of (unitary equivalence classes of) uniform Fredholm modules over the sums of (H ⊕ H, φ ⊕ 0), with homotopies also taken within this category (see 2.14). Fix (H, φ) for a time being, and let us define a homomorphism   ϕ0 : K1 Dφu (X) → K0u (X; φ) as follows: If U ∈ Mn (Dφu (X)) is a unitary representing a K1 -class, we set ϕ0 ([U ]) = [(H 2n , (φ ⊕ 0)n , U )]. It is immediate that (H 2n , (φ ⊕ 0)n , U ) is a uniform Fredholm module. Since homotopies of unitaries translate into operator homotopies of Fredholm modules and the operations on K1 and K0u are both direct sums, we see that ϕ0 is a group homomorphism. Analogously, we induce a homomorphism   ϕ1 : K0 Dφu (X) → K1u (X; φ) by assigning to a projection Q ∈ Mn (Dφu (X)) the triple (H 2n , (φ ⊕ 0)n , 2Q − 1). It is again easy to check that this triple is actually a uniform 1-Fredholm module. Since operations on K0 and K1u are both direct sums and homotopies translate to homotopies, we really do get a group homomorphism. Proposition 4.3. The above defined maps ϕ∗ : K1−∗ (Dφu (X)) → K∗u (X; φ) are isomorphisms.

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The proof follows the usual route of showing that elements of K∗u (X; φ) have nice representatives (cf. [8, Sections 8.3 and 8.4]). It is done in the following three lemmas. Lemma 4.4. Any element of K∗u (X; φ) may be represented by a uniform Fredholm module of the form (H 2n , (φ ⊕ 0)n , S), where S  1. Furthermore, homotopies can be also assumed to have this property. Proof. This is a standard cutting argument. We first deal  the even case. Take any repre with sentative (H 2n , (φ ⊕ 0)n , S). Consider the matrix S˜ = S0∗ 0S . It represents an odd selfadjoint operator in B(H 4n ), whose square differs from 1 by a uniform operator. Take the cutting function c : R → R given by ⎧ ⎨ −1, if t < −1, c(t) = t, if − 1  t  1, ⎩ 1, if t > 1. ˜ is again an odd selfadjoint operator (since c is odd), but with By functional calculus, c(S) ˜  1. Denote by T the upper right corner of c(S). ˜ Then T   1, and T − S is unic(S) form. The last statement can be seen by referring to the theorem on the essential spectrum of selfadjoint operators. The proof is completed by applying Lemma 2.16. The odd case is even more straightforward, since we may take a representative (H 2n , (φ ⊕ 0)n , P ) with P = P ∗ . Hence we can apply the cutting directly to P and replace it by c(P ). The same procedures can be applied to whole homotopies. 2 Lemma 4.5. Any element of K0u (X; φ) may be represented by a uniform 0-Fredholm module of the form (H 2n , (φ ⊕ 0)n , S), where S is a unitary. Furthermore, the homotopies can also be assumed to have this property. Proof. Take a representative (H 2n , (φ ⊕ 0)n , S), such that S  1. For simplicity, assume   12 , T , Sij ∈ B(H ). It follows that T   1, so the operator U = n = 1, so that S = ST21 SS22 √  T − 1−T T ∗  √ is well defined and unitary. 1−T ∗ T T∗ Since S is l-uniformly pseudolocal, T is l-uniformly pseudolocal and for any ε > 0, R, L  0 there exists M > 0, such that φ(f )S12 and S21 φ(f ) are (ε, M)-approximable for all f ∈ CR,L (X). Using this and uniformity of 1 − SS ∗ and 1 − S ∗ S, we conclude that 1 − T ∗ T and 1 − T T ∗ are uniform. Since Ψφ−1 (X) is a C∗ -algebra, so are their square roots. Consequently, S − U is uniform, and another application of Lemma 2.16 finishes the proof. Again, we can apply this procedure to the whole homotopy. 2 Lemma 4.6. Any class in K1u (X; φ) can be represented by a uniform 1-Fredholm module of the form (H 2n , (φ ⊕ 0)n , P ), where P 2 = 1. Proof. We proceed similarly as in the previous lemma. Choose a representative (H 2n , (φ ⊕ 0)n , P ), such that P = P ∗ and P   1. For simplicity, we assume that n = 1, and so   12 , where Q, Pij ∈ B(H ). It follows that Q is also selfadjoint and contractive. ThereP = PQ21 PP22 √  Q 1−Q2  is selfadjoint with O 2 = 1. fore, the operator O = √ 2 1−Q

−Q

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As in the previous proof, we obtain that 1 − Q2 is uniform and that P − O is uniform as well. This finishes the proof. 2 Let us now turn to relationship between K∗u (X, φ)’s for different φ’s. We shall need another definition (which is more general than what we need at the moment, but full generality will be required later): Definition 4.7. Let X and Z be spaces, let ϕ : C0 (X) → C0 (Z) be a ∗ -homomorphism, φX : C0 (X) → B(HX ) and φZ : C0 (Z) → B(HZ ) be ∗ -representations. We say that an isometry V : HZ → HX uniformly covers ϕ, if for every ε > 0, R, L  0 there exists M  0, such that V ∗ φX (f )V − φZ (ϕ(f )) is (ε, M)-approximable for every f ∈ CR,L (X). In short, V ∗ φX (·)V ∼lua φZ (ϕ(·)). We introduce a relation ≺ on the set X of (unitary equivalence classes of) ∗ -representations φ of C0 (X) on some (separable) Hilbert space, which turns it into a directed system. We define the relation ≺ by declaring that (H, φ) ≺ (E, ρ) (or just φ ≺ ρ) if and only if there exists an isometry Vφ,ρ : H → E which uniformly covers the identity map id : C0 (X) → C0 (X). The reflexivity of ≺ is obvious and the transitivity becomes clear after a momentary reflection on the definition of uniform covering. Furthermore, for φ, ρ ∈ X , we easily see that φ ≺ φ ⊕ ρ and ρ ≺ φ ⊕ ρ. If φ ≺ ρ, then we obtain a homomorphism iVφ,ρ : K∗u (X, φ) → K∗u (X, ρ) using Proposition 4.3 and the fact that Ad(Vφ,ρ ) maps Ψφ0 (X) into Ψρ0 (X) (where Ad(V ) is defined as Ad(V )(T ) = V T V ∗ and it is a ∗ -homomorphism when V is an isometry). This fact is a special case (when Z = X and π = id) of Lemma 5.4 from the next section, where we prove a more general statement requiring new notation. The set of K∗u (X, φ)’s, together with the maps iVφ,ρ , becomes a directed system indexed by X . The next lemma ensures that we may arbitrarily choose (and fix that choice of) an isometry Vφ,ρ for each pair φ ≺ ρ. Lemma 4.8. We adopt the notation from Definition 4.7. If two isometries V1 , V2 : HZ → HX uniformly cover ϕ, then the induced maps on K-theory are the same:         Ad(V1 ) ∗ = Ad(V2 ) ∗ : K∗ Ψφ0Z (Z) → K∗ Ψφ0X (X) . (Note that by the proof of Lemma 5.4, Ad(Vi )’s really map Ψφ0Z (Z) into Ψφ0X (X).) This lemma is analogous to the second part of [8, Lemma 5.2.4] and the proof carries over verbatim. This lemma also implies that ≺ becomes antisymmetric when it descends to K∗u (X, φ)’s. For each φ there is an obvious homomorphism jφ : K∗u (X; φ) → K∗u (X). It is also clear that jφ ’s commute with iVφ,ρ ’s, which allows us to state the final proposition of this section: Proposition 4.9. With the notation above,   K∗u (X) = lim jφ K∗u (X, φ) . φ∈X

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5. Mayer–Vietoris sequence The goal of this section is to prove the Mayer–Vietoris sequence for uniform K-homology groups: Theorem 5.1. Let A, B ⊂ X be closed subsets of X, such that A ∪ B = X, int(A ∩ B) = ∅ and d(A \ B, B \ A) > 0.4 Then there is a 6-term exact sequence K0u (A ∩ B)

K0u (A) ⊕ K0u (B)

K0u (X)

K1u (X)

K1u (A) ⊕ K1u (B)

K1u (X).

Before outlining the proof, we need a definition: Definition 5.2. Given a Hilbert space H and a ∗ -representation φ : C0 (X) → B(H ), we let Ψφ0 (X, Z) ⊂ Ψφ0 (X) to be the set of all operators T ∈ Ψφ0 (X) which are uniform on X \ Z, that is, such that for every ε > 0, R  0, there exists M > 0, such that for every f ∈ CR (X) with f |Z = 0 we have that φ(f )T and T φ(f ) are (ε, M)-approximable. Also, we set Dφu (X, Z) = 0 (X, Z) ⊂ B(H ⊕ H ). Ψφ⊕0 Note that a proof similar to the proof of Lemma 4.2 yields that Ψφ0 (X, Z) is a closed two-sided ideal of Ψφ0 (X). Proof of 5.1. The strategy is to first use the C∗ -algebra Mayer–Vietoris sequence (with φ fixed), and then apply Propositions 4.3, 4.9 and Excision Lemma 5.3 to obtain the result. Keeping the notation from 5.1, we have that Dφu (X, A) ∩ Dφu (X, B) = Dφu (X, A ∩ B) directly from the definitions, and Dφu (X, A) + Dφu (X, B) = Dφu (X) (by a partition of unity argument5 ). Subsequently, from the C∗ -algebra Mayer–Vietoris sequence, we get that K0 (Dφu (X, A ∩ B))

K0 (Dφu (X, A)) ⊕ K0 (Dφu (X, B))

K0 (Dφu (X))

K1 (Dφu (X))

K1 (Dφu (X, A)) ⊕ K1 (Dφu (X, B))

K1 (Dφu (X, A ∩ B))

is exact. The general Mayer–Vietoris sequence now follows by “taking the direct limit”, i.e. using naturality of our constructions, Proposition 4.9 and Excision Lemma 5.3. 2 4 This last condition just expresses the requirement that “the overlap of A and B does not get arbitrarily thin”. It is used only in the next footnote. 5 Take f, g ∈ C (X) with f + g = 1, f | b X\A = 0 and g|X\B = 0, f, g are L-continuous for some L (this is possible since d(A \ B, B \ A) > 0). Write T = T φ(f ) + T φ(g). Now if h|A = 0, then T φ(f )φ(h) = 0 and φ(h)T φ(f ) = [φ(h), T ]φ(f ) + T φ(h)φ(f ).

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It remains to deal with the excision lemma. For the rest of this section, we shall denote by X a proper metric space, by Z ⊆ X a closed subset of X and by φ : C0 (X) → C0 (Z) the restriction homomorphism. Lemma 5.3 (Excision lemma). There is a natural isomorphism     lim K∗ Dφu (X, Z) ∼ = lim K∗ DφuZ (Z) . φ

φZ

By virtue of 4.9, we may say that the “relative uniform K-homology” K∗u (X, Z) is isomorphic to K∗u (Z). Proof. The strategy is obtain a commutative diagram (notation will be introduced in the course of the proof)

Ad(V )

K∗ (Ψφ0X (X, Z)) Ad(W V )

K∗ (Ψφ0Z (Z))

Ad(SW ) Ad(W )

Ad(S)

K∗ (Ψφ0 (X, Z)) X

(1)

K∗ (Ψφ0 (Z)) Z

starting with the following data: a representation φX : C0 (X) → B(HX ), a representation φZ : C0 (Z) → B(HZ ) and an isometry V : HZ → HX , which uniformly covers π (this gives the first  in (1)). In the diagram, the horizontal arrows shall uniformly cover the identity (on the level of K-theory), and the diagonals heading up will uniformly cover π . This would establish the lemma. Let us now explain how can we arrange the starting data. If we start with a ∗-representation φX : C0 (X) → B(HX ), it induces a Borel measure on X, and extends to a ∗-representation (also denoted by φX ) of ∞ (X). In particular, we may restrict φX to a representation φZ : C0 (Z) → B(HX ) and let HZ = χZ HX . Then the inclusion V : HZ → HX actually exactly covers π , i.e. V ∗ φX (f )V = φZ (π(f )) = φ(χZ f ) for all f ∈ C0 (X). Conversely, starting with a ∗-representation φZ : C0 (Z) → B(HZ ), we obtain a ∗-representation φX = φZ ◦ π of C0 (X), so that we can put HX = HZ and V = id. The rest of the proof is devoted to obtaining a diagram (1) from given φX , φZ and V uniformly covering π . We accomplish our goal similarly as [8, Proof of 3.5.7]. Lemma 5.4. Let φX : C0 (X) → B(HX ), φZ : C0 (Z) → B(HZ ) be ∗ -representations and let V : HZ → HX be an isometry which uniformly covers π . Then   Ad(V ) Ψφ0Z (Z) ⊂ Ψφ0X (X, Z). (The adjoint map Ad is defined as Ad(V )(T ) = V T V ∗ , and it is a ∗ -homomorphism since V is an isometry.) Proof. We first show that V V ∗ ∈ Ψφ0X (X, Z). Decompose HX = V V ∗ HX ⊕(1−V V ∗ )HX . With    φ φ12  . The fact that V V ∗ is respect to this decomposition V V ∗ = 10 00 , and we denote φX = φ11 21 φ22 φX -uniformly pseudolocal is equivalent to φ12 (·)

and φ21 (·)

are l-uniformly approximable.

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Using the covering assumption,   φ11 (f ∗ f ) = V V ∗ φX (f ∗ f )V V ∗ ∼lua V φZ π(f ∗ f ) V ∗  ∗   = V φZ π(f ) φZ π(f ) V ∗ ∼lua V V ∗ φX (f )∗ V V ∗ φX (f )V V ∗ = φ11 (f )∗ φ11 (f ). Since φX is a ∗ -homomorphism, we have φ21 (f )∗ φ21 (f ) = φ11 (f ∗ f ) − φ11 (f )∗ φ11 (f )

(2)

for each f ∈ C0 (X). In other words, φ21 (·)∗ φ21 (·) is l-uniformly approximable. Using the spec√ tral theorem for compact selfadjoint operators,6 also φ21 (·)∗ φ21 (·) = |φ21 (·)| is l-uniformly approximable. Let φ21 (f ) = u(f )|φ21 (f )| denote the polar decomposition. From this formula, it follows that φ21 (f ) is l-uniformly approximable as well. To show that V V ∗ is uniform on X \ Z, it suffices to note that in addition to φ12 (·) and φ21 (·) being l-uniformly approximable, we also have φ11 (f ) = V V ∗ φX (f )V V ∗ ∼lua V φZ (π(f ))V ∗ = 0 for f ∈ C0 (X \ Z). We have shown that V V ∗ ∈ Ψφ0X (X, Z). From this, we easily get that Ad(V ) maps Ψφ0Z (Z) into Ψφ∗X (X, Z). 2 Let σ : C0 (Z) → C0 (X) be a completely positive lift of π that satisfies • if f ∈ CR (X) then supp(σ (π(f ))) ⊂ {x ∈ X | d(x, supp(f ))  1}, • there exists L , such that if f is L-continuous then σ (f ) is L + L -continuous. In particular, if g ∈ CR (Z) then σ (g) ∈ CR+2 (X). Such a lift exists.7 Now φX σ : C0 (Z) → B(HX ) is a completely positive map, so by the Stinespring’s theorem, there exist a Hilbert space H and maps ρ12 , ρ21 , ρ22 such that φZ =



φX σ ρ21

ρ12 ρ22

 : C0 (Z) → B(HX ⊕ H )

is a ∗ -homomorphism. Denote by W : HX → HX ⊕ H the obvious inclusion. Claim 2. Ad(W ) maps Ψφ0X (X, Z) into Ψφ0 (Z). Furthermore W V uniformly covers id: Z

C0 (Z) → C0 (Z). In other words, W ∗ V ∗ φZ (·)V W − φZ (·) is l-uniformly approximable on C0 (Z). Proof. Decomposing into matrices shows that Ad(W )(T ) belongs to Ψφ0 (Z) if and only if Z

both T ρ12 (·) and ρ21 (·)T are l-uniformly approximable. Since φZ is a ∗ -homomorphism, ∗ (f )ρ (f )T is l-uniformly ρ21 (f )∗ ρ21 (f ) ∈ φX (C0 (X \Z)) for all f ∈ C0 (Z), cf. (2). Hence ρ21 21

6 If k ∈ K is selfadjoint, then for ε > 0 we can approximate k by a rank-M operator, where M is the sum of dimensions of eigenspaces corresponding to all eigenvalues λ with |λ| > ε. 7 Note that a positive map between commutative C∗ -algebras is automatically completely positive, and a nice positive linear lift can be constructed using a linear basis and the Urysohn lemma–type construction. The L-continuity can be also arranged.

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approximable. Consequently, (ρ21 (f )T )∗ (ρ21 (f )T ) is l-uniformly approximable as well, and it follows by the argument in the proof of Lemma 5.4 that ρ21 (f )T itself is as well. This finishes the first part. To see that W V uniformly covers id on C0 (Z), just observe that for f ∈ C0 (Z), we have V ∗ W ∗ φZ (f )W V = V ∗ φX (σ (f ))V ∼lua φZ (π(σ (f ))) = φZ (f ) by assumption of V . 2 The next step is to consider the Hilbert space HX = HX ⊕ (HX ⊕ H ) with the ∗ -representation = φX ⊕ φZ π of C0 (X). Denote by S : HX ⊕ H → HX the inclusion (HX is included as the second HX summand).

 φX

Claim 3. S uniformly covers π . Ad(SW ) is homotopic to a ∗ -homomorphism which uniformly covers id : C0 (X) → C0 (X). Hence we are in the position to iterate the construction we have done so far to obtain a commutative diagram (1).  S = φ  π . Continuing with the second Proof. In fact, S actually exactly covers π , since S ∗ φX Z part of the claim, note that SW includes HX into HX ⊕ HX ⊕ H as the second copy of HX . If we denote by Y : HX → HX ⊕ HX ⊕ H the inclusion as the first summand, then Ad(Y ) exactly covers id : C0 (X) → C0 (X). Furthermore, Ad(SW ) and Ad(Y ) are homotopic via the homotopy of ∗ -homomorphisms

⎛

sin2 ( π2 t)T

sin( π2 t) cos( π2 t)T

At : T → ⎝ sin( π2 t) cos( π2 t)T 0

cos2 ( π2 t)T 0

0

⎞

0⎠, 0

t ∈ [0, 1].

It remains to verify that At maps Ψφ0X (X, Z) into Ψφ0 (X, Z). To this end, it is enough to observe X  T T 0 0  -l-uniformly pseudolocal ˜ that if T ∈ ΨφX (X, Z), then T = T T 0 ∈ B(HX ⊕ HX ⊕ H ) is φX 0 0 0

and uniform on C0 (X \ Z). For f ∈ C0 (X), we compute ⎛ T φX σ π(f ) − φX (f )T T φX (f ) − φX (f )T   T˜ , φX (f ) = ⎝ T φX (f ) − φX σ π(f )T T φX σ π(f ) − φX σ π(f )T 0 0

⎞ 0 0⎠. 0

It is now clear that for showing l-pseudolocality of T˜ it suffices to see that T φX (f ) − φX σ π(f )T = [T , φX (f )] + (φX (f − σ π(f )))T is l-uniformly approximable. But f − σ π(f ) ∈ C0 (X \ Z), hence the assertion follows from the assumptions on T and the lift σ .  T φX (f ) T φX σ π(f ) 0   (f ) = T φ (f ) T φ σ π(f ) 0 and the uniformness of T˜ on C (X \ Z) follows Similarly T˜ φX X X 0 0

0

0

from the observation that π(f ) = 0 for f ∈ C0 (X \ Z). This finishes the proof of Lemma 5.3.

2

2

6. Coarse geometry and C∗ -algebras The first part of this section is devoted to a review of basic notions from coarse geometry. The second part recalls the definitions of C∗ -algebras reflecting the coarse structure: (uniform) Roe C∗ -algebras.

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Coarse geometry studies large-scale behavior of spaces. While it is possible to give an abstract definition of a coarse structure (see [13]), for our purposes it is sufficient and more straightforward to assume that our spaces are endowed with a metric. The appropriate notion of maps in the “coarse category” is the following: Definition 6.1. A (not necessarily continuous) map g : X → Z between metric spaces X and Z is said to be coarse, if: • For any r  0 there exists R  0, such that d(x1 , x2 )  r implies d(g(x1 ), g(x2 ))  R for x1,2 ∈ X. An equivalent condition is that there exists a non-decreasing function ρ + : R+ → R+ , such that d(g(x1 ), g(x2 ))  ρ + (d(x1 , x2 )). • For any r  0 we have diam(g −1 (B(z, r))) < ∞ for all z ∈ Z. This condition is referred to as being cobounded. Furthermore, we say that g is called uniformly cobounded, if for any r  0, we have    Rg (r) := sup diam g −1 B(z, r) < ∞. z∈Z

When working in the “coarse category”, we may choose a nice representative in the class of coarsely equivalent spaces: Definition 6.2. A metric space Y is said to be uniformly discrete, if there is δ > 0, such that d(x, y)  δ whenever x = y ∈ Y . Furthermore, Y is said to have bounded geometry, if for any r  0 we have   supB(y, r) < ∞. y∈Y

When switching between discrete and “continuous spaces”, the following concept proved to be useful: Definition 6.3. Let X be a metric space and let d  0. The Rips complex Pd (X) is a simplicial polyhedron defined as follows: • the vertex set of Pd (X) is X, • any q + 1 vertices x0 , x1 , . . . , xq span a simplex of Pd (X) if and only if d(xi , xj )  d,

∀i, j ∈ {0, . . . , q}.

Note that if X has bounded geometry, Pd (X) is locally finite and finite dimensional. We endow it with the geodesic metric. We now define C∗ -algebras, which reflect large-scale behavior of metric spaces. Let Y be a uniformly discrete metric space with bounded geometry. We consider the Hilbert space 2 (Y ) ⊗ 2 (N) ∼ = 2 (Y × N) (or 2 (Y )), and represent bounded operators T on it as matrices T = (tyx )x,y∈Y with entries tyx in B(2 (N)) (or C respectively).

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Definition 6.4 (Finite propagation: Discrete version). We say that an operator T = (tyx ) ∈ B(2 (Y × N)) (or B(2 (Y ))) has finite propagation, if there exists R  0, such that tyx = 0 whenever d(x, y) > R. The smallest such R is called the propagation of T and denoted by propagation(T ). Definition 6.5. We say that T is locally compact, if tyx ∈ K (2 (N)) for all x, y ∈ Y . (This condition is void in the case T ∈ B(2 (Y )).) We say that T has uniformly bounded coefficients, if there exists C > 0, such that tyx   C for all x, y ∈ Y . Definition 6.6. The norm-closure of the algebra of all finite propagation operators with uniformly bounded coefficients in B(2 (Y )) is said to be the uniform Roe C∗ -algebra of Y , denoted by Cu∗ Y . We denote by Ck∗ (Y ) the norm-closure of the algebra of all locally compact finite propagation operators T = (tyx ) with uniformly bounded coefficients in B(2 (Y × N)), which satisfy the additional condition that the set {tyx | x, y ∈ Y } ⊂ K (2 (N)) is compact in the norm topology on K (2 (N)). Remark 6.7. The additional condition in the previous definition merely says that up to ε, we have only finitely many entries tyx . Another way of stating this condition is that for each ε > 0 there exists M  0, such that each txy , x, y ∈ Y , is at distance at most ε from a rank-M operator. Remark 6.8. The C∗ -algebra Cu∗ Y is not functorial under coarse uniformly cobounded maps, as an examples of one-point and two-point spaces show. Nevertheless, coarsely equivalent spaces have Morita equivalent uniform Roe C∗ -algebras, see [5]. This corresponds to the fact that Ck∗ (Y ) is functorial under such maps. We now cite a proposition, which provides an estimate on the norm of an operator in terms of its entries: Proposition 6.9. (See [13].) Let Y be a uniformly discrete space with bounded geometry, and let t = (tyz )y,z∈Y be a matrix with entries tyz ∈ K (H ) [or tyz ∈ C]. For every P > 0 there is C > 0, such that if t has propagation at most P , we have t  C supy,z tyz , with the operator norm in B(2 Y ⊗ H ) [or B(2 Y ) respectively]. To finish the section, we show that as far as K-theory of uniform Roe algebras is concerned, we may work with Ck∗ (Y ). Lemma 6.10. Let Y be a uniformly discrete metric space with bounded geometry. Then Ck∗ (Y ) ∼ = Cu∗ Y ⊗ K . Proof. We show that Cu∗ Y ⊗ K (2 (N)) is dense in Ck∗ (Y ) (with the obvious inclusion). Pick T = (tyx ) ∈ Ck∗ (Y ) and > 0. Denote the propagation of T by p. By Proposition 6.9, there is a constant C > 0, such that if S = (syx ) is a matrix of compacts with propagation at most p, then S  C supx,y∈X syx . Since {tyx | x, y ∈ Y } is compact, there is an /C-net t1 , . . . , tm in it. Then clearly T is -far from an operator of the form T1 ⊗ t1 + · · · + Tm ⊗ tm , where each Ti ∈ Cu∗ Y . This shows the density, which implies that Ck∗ (Y ) and Cu∗ Y ⊗ K (2 (N)) are actually isomorphic. 2

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7. Finite propagation representatives In this section, we prove that any class in a uniform K-homology group can be represented by a uniform Fredholm module with the operator having finite propagation. The proof follows the outline of the proof of analogous result in analytic K-homology. Definition 7.1. An open cover of X is said to • have finite multiplicity, if for any R  0 there is K  0, such that any ball with radius R intersects at most K elements of the cover; • be uniformly bounded, if there is a common upper bound for all the diameters of members of the cover. Remark 7.2. Any space X with bounded geometry admits uniformly bounded covers with finite multiplicity. However, bounded geometry alone produces such covers with possibly large bound on the diameters of the cover members. Consequently, a priori the propagation might not be made arbitrarily small (see the proof the next proposition). In order to achieve small propagation, we need some small scale (topological) assumption; for instance finite covering dimension would suffice. Definition 7.3 (Finite propagation: Continuous version). Let H be a Hilbert space and let φ : C0 (X) → B(H ) be a ∗ -representation. We say that T ∈ B(H ) has finite propagation, if there exists R > 0, such that φ(f )T φ(g) = 0 for every f, g ∈ C0 (X) with d(supp(f ), supp(g))  R. Proposition 7.4. Each uniform K-homology element over a space X with bounded geometry can be represented by a uniform Fredholm module (H, φ, S), where S is a finite propagation operator. Furthermore, we may assume that homotopies go through finite propagation operators as well. Proof. Let (H, φ, T ) be a uniform Fredholm module. Take a uniformly bounded open cover (Ui )i∈I with finite multiplicity, and let (ϕi2 )i∈I be a continuous partition of unity subordinate to (Ui )i∈I . By replacing the sets Ui by Nδ (Ui ), the δ-neighborhoods for a fixed δ > 0 and obtaining a partition of unity for the cover (Nδ (Ui ))i , we can assume that all ϕi ’s are L0 -continuous for some L0  0.  Denote S = i∈I ϕi T ϕi . This operator has finite propagation (which is bounded from above by supi diam(Ui )). We prove that (H, φ, S) is a uniform Fredholm module which represents the same uniform K-homology element as (H, φ, T ). Fix ε > 0 and R, L > 0. Let M be such that [T , φ(·)] is (ε, R, 2 max(L0 , L), M; φ)approximable and that T φ(·) and φ(·)T are (ε, R, M; φ)-approximable. Denote S  = S − T =  i∈I ϕi [T , ϕi ]. By finite multiplicity assumption, there is M1 , such that  any ball with radius R intersects at most M1 sets Ui . Take f ∈ CR (X) and consider f S  = i f ϕi [T , ϕi ]. This sum has at most M1 nonzero terms, and each of them is (ε, M)-approximable, hence f S  itself is (M1 ε, MM1 )-approximable. Similarly for f ∈ CR,L (X),





ϕi [T , ϕi ]f = ϕi T ϕi f − ϕi2 Tf = (ϕi T ϕi f − ϕi2 f T ) + ϕi2 [f, T ] Sf = i

=

i

i

ϕi [T , f ϕi ] + [f, T ].

i

i

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The last term is (ε, M)-approximable by assumption, and again only at most M1 terms in the sum are nonzero, and all of them are (ε, M)-approximable. Consequently, S  f is (M1 + 1)ε, MM1 + 1)-approximable. Therefore we have proved that S  is uniform. Applying Lemma 2.16 finishes the first part of the proof.  For the part on homotopies, we just need to observe that the formula i∈I ϕi T ϕi produces a continuous family if we vary T continuously, thanks to finite multiplicity of the chosen cover. 2 8. Another picture of uniform Roe algebras The definition of Ck∗ (Y ) as given in Section 6 inherently uses the standard basis of the auxiliary Hilbert space 2 N. In this section, we develop a picture of Ck∗ (Y ) starting with a general X-module (H, φ), instead of the concrete one (2 Y ⊗ 2 N with the multiplication action). Furthermore, this model allows us to translate from “continuous” spaces X (which are needed in order to observe more than just 0-dimensional phenomena in (uniform) K-homology) to their discrete models Y ⊂ X (which are supposed to be the targets of the index/assembly map). Let us fix a metric space X for the rest of this section. Definition 8.1. We say that Y ⊂ X is a quasi-lattice, if Y with induced metric is uniformly discrete space with bounded geometry, which is coarsely equivalent to X. We say that a collection (Vy )y∈Y of subsets of X is a quasi-latticing partition, if each Vy  is open, Vx ∩ Vy = ∅ if x = y, X = y∈Y Vy , supy∈Y diam(Vy ) < ∞ and for every ε > 0, supy∈Y |{z ∈ Y | Vz ∩ Nbhdε (Vy ) = ∅}| < ∞. Remark 8.2. Not all spaces X have a quasi-lattice, but those with “bounded geometry” in any reasonable sense do. Furthermore, once there is a quasi-lattice, it is easy to produce quasilatticing partitions (for instance by means of “pick the closest point in Y ” map). Example 8.3. A useful example to have in mind is the one of a graph X (with edges attached), with Y being its 0-skeleton. More generally, 0-skeleton of a uniformly locally finite simplicial polyhedron (endowed with the geodesic metric) is a quasi-lattice. Recall that any ∗ -homomorphism φ : C0 (X) → B(H ) induces a Borel measure on X, and extends to a representation (also denoted by φ) of ∞ (X). We shall use this fact without mentioning explicitly throughout this section. Definition 8.4 (Bases choice). Given a metric space X, we define the bases choice A for X to be a 5-tuple (Y, (Vy )y∈Y , H, φ, {Sy }y∈Y ), where • • • •

Y ⊂ X is a quasi-lattice of X, (Vy )y∈Y is a quasi-latticing partition of X, H is a Hilbert space, φ : C0 (X) → B(H ) a non-degenerate ∗-representation,8 y Ny Sy = (ei )i=1 is a basis of Hy = φ(χVy )H (where we allow Ny ∈ N ∪ {∞} and we put by convention that Sy = ∅ if Hy = {0}).

8 A representation φ : C (X) → B(H ) is non-degenerate, if [φ(C (X))]⊥ = {0} 0 0

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Such a bases choice determines a (possibly non-surjective) isometry uA : H = 2 (Y × N).

 y

Hy →

Definition 8.5. Let X be a metric space, Y ⊂ X a quasi-lattice, (Vy )y∈Y a quasi-latticing partition, and let Ai = (Y, (Vy )y∈Y , Hi , φi , {Syi }y∈Y ), i = 1, . . . , k be bases choices. Define the  C∗ -algebra Ck∗ (X, A1 , . . . , Ak ) ⊂ B( ki=1 Hi ) as the closure of the algebra of the operators k T ∈ B( i=1 Hi ) satisfying the following conditions: • T has finite propagation, • there exists M  0, such that each “entry” Tj,i;y,x : φi (χVx )Hi → φj (χVy )Hj only uses the j

first M basis vectors from bases Sxi , Sy . There is an injective ∗ -homomorphism   Ad(uA1 ⊕ · · · ⊕ uAk ) : Ck∗ (X, A1 , . . . , Ak ) → Mk Ck∗ (Y ) . We call the C∗ -algebra Ck∗ (X, A ) ⊂ B(H ) the A -realization of Ck∗ (Y ). Remark 8.6. Note that Ck∗ (X, A ) is isomorphic only to a subalgebra of Ck∗ (Y ) in general, but if each Sy is infinite, then Ck∗ (Y ) and Ck∗ (X, A ) are isomorphic. Define supp(A ) = {y ∈ Y | Sy = ∅}. If supp(A ) is coarsely equivalent to Y , we have that K∗ (Ck∗ (Y )) ∼ = K∗ (Ck∗ (X, A )). More precisely, Ck∗ (Y ) and Ck∗ (X, A ) are Morita equivalent. Indeed, M∞ (Ck∗ (X, A )) ∼ = M∞ (Cu∗ (supp(A ))), for Morita equivalence of Cu∗ (supp(A )) and Cu∗ Y we refer to [5]. We continue by defining a relation between tuples of bases choices, in order to be able to get an inductive limit of realizations of Ck∗ (Y ). We begin by a notion similar to an inclusion between a pair of bases choices.   Definition 8.7. Fix a quasi-lattice Y ⊂ X. Let Ai = Y, (Vyi )y∈Y , Hi , φi , {Syi } , i = 1, 2, be bases choices. We shall write A1 ⊆ A2 , if the following conditions are satisfied: • For each y ∈ Y , φ(χVy1 )H1 is isometric to a subspace of φ(χVy2 )H2 via an isometry vy . • Each vy maps nth vector in the basis Sy1 to the n-vector in the basis Sy2 . A weaker version of ⊆, denoted now by A1  A2 , is defined in the same manner, except the last condition is replaced by • for all k ∈ N there is l ∈ N, such that for all y ∈ Y the vy -images of the first k vectors of Sy1 are among the linear span of the first l vectors of Sy2 . We now extend this inclusion to lists. Given two lists of bases choices (A1 , . . . , Ak ) and (A1 , . . . , Al ) for X with respect to Y , we declare (A1 , . . . , Al ) ≺ (A1 , . . . , Ak ), if there is an

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injective function σ : {1, . . . , l} → {1, . . . , k}, such that Ai ⊆ Aσ (i) for all i = 1, . . . , l. If this happens, then there is a natural embedding   i : Ck∗ X, A1 , . . . , Al → Ck∗ (X, A1 , . . . , Ak )  (implemented by the ∗ -homomorphism Ad(V ), where V = y vy is the isometric embedding of appropriate Hilbert spaces). This embedding commutes with maps between matrix algebras over Ck∗ (Y ) as follows: Ck∗ (X, A1 , . . . , Ak )

Ck∗ (X, A1 , . . . , Al ) Ad(uA  ⊕···⊕uA  )

Ad(uA1 ⊕···⊕uAk )

l

1

Ml (Ck∗ (Y ))

hσ ⊗id

(3)

Mk (Ck∗ (Y )).

By hσ : Ml (C) → Mk (C) we denote the embedding of matrix algebras determined by σ . More precisely, hσ is the linear extension of the following assignment of matrix units Ml (C)  eij → eσ (i)σ (j ) ∈ Mk (C). Furthermore, if we assume that Ai = Aσ (i) for i = 1, . . . , l, and if supp(Aj ) is coarsely equivalent to Y for each j = 1, . . . , k, then the top horizontal map induces an isomorphism on K-theory. This is a straightforward generalization of Remark 8.6. Note that for any bases choice A = (Y, (Vy )y∈Y , H, φ, {Sy }y∈Y ), there is another one A  with A ⊆ A  , such that supp(A  ) = Y . This can be arranged by choosing the Hilbert space of A  to be H  = H ⊕ 2 (Y × N), the direct sum action of C0 (X) and a suitable choice of bases Sy . The previous discussion, together with Lemma 6.10, culminates in the following proposition: Proposition 8.8. Let X be metric space and let Y ⊂ X be a quasi-lattice. The collection X of all finite lists (A1 , . . . , Ak ) of bases choices for X with Y fixed forms a directed system. We have that there is an isomorphism η    ∼  = η : lim K∗ Ck∗ (X, A1 , . . . , Ak ) −→ K∗ Cu∗ Y . X

The following lemma shows that given a finite propagation uniform operator T on an Xmodule H , we can always find a bases choice A , such that T ∈ Ck∗ (X, A ). Lemma 8.9. Let X be metric space, let Y ⊂ X be a quasi-lattice and let (Vy )y∈Y be a quasi-latticing partition of X. Let H be a Hilbert space and let φ : C0 (X) → B(H ) be a ∗ homomorphism. Given a finite collection T1 , . . . , Tk ∈ B(H ) of uniform operators with finite propagation, there exists a bases choice A , such that Ti ∈ Ck∗ (X, A ) for all i = 1, . . . , k. Proof. For simplicity, assume that we are given just one T ∈ B(H ) to deal with (it will be clear that we can follow the procedure outlined below simultaneously for finitely many operators). Denote Hy = φ(χVy H ) and Tyz = φ(χVy )T φ(χVz ) ∈ B(Hz , Hy ). Since T has finite propagation and Y is uniformly discrete, there is a K, such that there are at most K nonzero entries in each column and row of the matrix (Txz )x,z∈Y .

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Fix ε1 = 1 and take R > supy∈Y diam(Vy ). It follows from the assumption that there exists M, such that each Tyz is (ε1 , M)-approximable. Therefore, for each y ∈ Y , there are 2M orthonormal y y vectors e1 , . . . , e2M ∈ Hy , for which there are 2M × 2M-matrices which in these (partial) bases represent operators sy ∈ B(Hy ) with Tyy − sy  < ε1 . Fix y ∈ Y for a while and consider the “column” (Tyz )z∈Y . Each of them is (ε1 , M)y y approximable, but not necessarily by a matrix in the partial basis e1 , . . . , e2M chosen so far. By adding at most M vectors to the chosen partial bases for Hy and Hz respectively, we can ensure that Tyz will be (ε1 , M)-approximable in the partial bases of Hy and Hz . We can do this for each nonzero Tyz , z ∈ Y , resulting in having chosen partial basis for Hy having at most (2 + K)M elements, and partial bases for Hz ’s having at most 3M elements. Doing this process for all y ∈ Y results in choosing partial bases for each Hy having at most (2 + 2K)M elements, now with the property that each Tyz is (ε1 , M)-approximable with matrices in the chosen partial bases. To finish the construction, we choose a sequence of εn > 0 converging to 0 and do the above described process for each n, always just adding the newly chosen partial bases to the previous ones. Hence, we have constructed A = A (Y ). The fact that T ∈ Ck∗ (X, A ) follows easily from the construction and the estimate 6.9. 2 In fact, we can improve the previous lemma to finite collections of uniform operators which are do not necessarily have finite propagation, but are only approximable by finite propagation ones. To carry out the argument, we are going to use the relation  on bases choices (see Definition 8.7). Note that if A1  A2 , then Ck∗ (X, A1 ) ⊂ Ck∗ (X, A2 ): Let w ∈ Ck∗ (X, A1 ) be finite propagation operator, such that a bound M on the number of basis vectors from Sy1 which are used in each entry wyz of w. By the last condition in the definition of , there is a number M  , such that for each y ∈ Y , the first M vectors of Sy1 are in the linear span of the first M  vectors of Sy2 . Consequently, entries wyz use only the first M  vectors of bases Sy2 , and so w ∈ Ck∗ (X, A2 ). Lemma 8.10. Let X be metric space, let Y ⊂ X be a quasi-lattice and let (Vy )y∈Y be a quasi-latticing partition of X. Let H be a Hilbert space and let φ : C0 (X) → B(H ) be a ∗ -homomorphism. Given a finite collection T , . . . , T in Θ(φ), the C∗ -algebra generated by uni1 k form operators with finite propagation, there exists a bases choice A , such that Ti ∈ Ck∗ (X, A ) for all i = 1, . . . , k. We isolate a part of the proof of the above lemma as another lemma, as it is useful by itself. Lemma 8.11. Let X be metric space, let Y ⊂ X be a quasi-lattice and let (Vy )y∈Y be a quasi-latticing partition of X. Let H be a Hilbert space and let φ : C0 (X) → B(H ) be a ∗ -homomorphism. Assume that we are given a countable collection A , . . . , A , . . . of bases 1 n choices of the form (Y, (Vy )y∈Y , H, φ, ·). Then there exists a bases choice A of the same form, such that Ai  A , i  1. Proof. Denote An = (Y, (Vy )y∈Y , H, φ, {Syn }y∈Y ). We now define bases Sy out of Syn (and put A = (Y, (Vy )y∈Y , H, φ, {Sy }y∈Y )). Fix y ∈ Y and enumerate the orthonormal bases Syn of the Hilbert space φ(χVy )H as (ein )i1 . We make one basis out of this sequence as follows: we fix a bijection α : N × N → N (for instance α(n, i) = 12 (n + i − 1)(n + i − 2) + i; say we think of N × N to be the lattice points in the first quadrant of the plane, and we enumerate the points along

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the diagonals going from “top-left” to “right-bottom”). Let (β1 , β2 ) : N → N × N be its inverse. β (k) Now take the sequence of vectors k → eβ21(k) , and apply the Gramm–Schmidt orthogonalization process to it. We obtain a new basis Sy , which obviously has the following property: for each n  1 and i  1, the vectors ein , . . . , ein are in the linear span of the first α(n, i) basis vectors of the new basis. A quick glance at the definition of the relation  for bases choices shows that A is as required. 2 Proof of Lemma 8.10. For simplicity, we concentrate on the case that k = 1, i.e. when we are given one operator T ∈ Θ(φ). Note that T is uniform by the argument of Lemma 4.2. By assumption, T is approximable by a sequence Tn of uniform operators with finite propagation. For each Tn , there is a bases choice An = (Y, (Vy )y∈Y , H, φ, {Syn }y∈Y ), such that Tn ∈ Ck∗ (X, An ). Applying the previous lemma yields a bases choice A , such that An  A for each n  1. Since Ck∗ (X, An ) ⊂ Ck∗ (X, A ), Tn is a sequence of operators in Ck∗ (X, A ) which converges to T . This finishes the proof. 2 9. The uniform index map In the usual analytic K-homology, there is the index map (often also called the coarse assembly map) from the K-homology K∗ (X) of a space X to the K-theory of its Roe algebra K∗ (C ∗ X). But since Roe algebras of coarsely equivalent spaces are isomorphic, the target group of the index map can be understood as the K-theory K∗ (C ∗ Y ) of the Roe algebra of any quasilattice Y ⊂ X. The quickest way to define this map in the usual case is to use the reformulation of the Khomology as K-theory of a dual algebra (see [8, Theorem 8.4.3] and Section 4 for an analogous result in the uniform case) and then the 6-term exact sequence in K-theory, whose boundary maps become the assembly maps. For details of this construction, see for instance [8, Section 12.3]. The goal of this section is to construct a similar index/assembly map in the uniform setting. More precisely, we define a homomorphism μu : K∗u (X) → K∗ (Cu∗ Y ) for a quasi-lattice Y ⊂ X in a metric space X. However, instead of the C∗ -algebra route, we take a more hands-on approach. In this paragraph, we recall a formula for the usual assembly map. If (H, φ, S) is a 0-Fredholm module, we can define its index as follows: denote  W=

1 S 0 1



1 −S ∗

0 1



1 0

S 1



0 1

This is an invertible in M2 (B(H )). Then put ind(S) = W  ind(S) =

SS ∗ + (1 − SS ∗ )SS ∗ S ∗ (1 − SS ∗ )

−1 0



  ∈ M2 B(H ) .

 1 0  −1 W ∈ M2 (B(H )). Concretely, 00

 S(1 − S ∗ S) + (1 − SS ∗ )S(1 − S ∗ S) . (1 − S ∗ S)2

A simple computation shows  that ind(S) is actually an idempotent in M2 (B(H )). Furthermore, ∂(H, φ, S) = [ind(S)] − [ 10 00 ] is a K0 -class in the K-theory group of appropriate algebra, modulo which is S invertible. For example, starting with a finite propagation S, one gets ∂(H, φ, S) in K0 (C ∗ X), the K-theory of the Roe C∗ -algebra.

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Starting with a 1-Fredholm module (H, φ, Q), its index can be constructed using the formula 9 ind(Q) = exp(−2πi Q+1 2 ) ∈ B(H ). The operator ind(Q) is invertible, but even if we start with a finite propagation Q, ind(Q) might not have finite propagation. However, it is approximable by finite propagation invertibles in this case, hence still gives a class [ind(Q)] ∈ K1 (C ∗ X). Let us now turn to the uniform case. Fix a quasi-lattice Y ⊂ X. We define μu : K∗u (X) → K∗ (Cu∗ Y ) in the following proposition: Proposition 9.1 (Uniform index map, even case). Let (H, φ, S) be a 0-uniform Fredholm module with S having finite propagation and φ being non-degenerate. For any quasi-lattice Y ⊂ X, there y exists a bases choice A = (Y, (Vy )y∈Y , H, φ, {(ei )i∈N }y∈Y ), such that ind(S) ∈ M2 (B(H )) is an idempotent that actually belongs to Ck∗ (X, A , A ). Furthermore, we can define a group homomorphism μu : K0u (X) → K0 (Cu∗ Y ) by        1 0 ∈ K0 Cu∗ Y , μu (H, φ, S) = η∗ ind(S) − 0 0 i.e. the right-hand side does not depend on the choices made. Recall that η is described in Proposition 8.8. Proposition 9.2 (Uniform index map, odd case). Let (H, φ, Q) be a 1-uniform Fredholm module with Q having finite propagation and φ being non-degenerate. For any quasi-lattice Y ⊂ X there y exists a bases choice A = (Y, (Vy )y∈Y , H, φ, {(ei )i∈N }y∈Y ), such that ind(Q) ∈ B(H ) is an invertible that actually belongs to Ck∗ (X, A )+ . Furthermore, the map μu : K1u (X) → K1 (Cu∗ Y ) defined by    μu [H, φ, Q] = η∗ ind(Q) ∈ K1 Cu∗ Y is a group homomorphism. Proof of the 0-case. Picking any quasi-latticing partition (Vy )y∈Y , the existence of a suitable A follows from Lemma 8.9, applied to the four entries of ind(S), which are uniform and have finite propagation. It is clear that our construction of the index preserves direct sums. Also, the index of a degenerate element gives zero in Indeed, if (H, φ, S) is a degenerate 0-Fredholm   the) 0K-theory. for any f ∈ C0 (X), so by using a partition of unity we module, then φ(f ) ind(S) = φ(f 0 0   obtain that ind(S) = 10 00 . Thus, to finish the proof, we need to show the independence of the index on the choice of A , and under homotopies of uniform Fredholm modules. Our proof for homotopies includes the argument for choices of A , since we can just take a constant homotopy, and choose different bases choices at the endpoints. We shall now outline the proof for homotopies. Assume that we are given a homotopy (H, φt , St ) of uniform Fredholm modules. We assume that all St have finite propagation (see proposition 7.4), so that the index as we have defined it can be constructed. Note that the requirements on φt ensure that B = Θ(φt ), the C∗ -algebra generated by all φ-uniform operators with φ-finite propagation, does not depend on t. 9 When we talk about invertibles in a non-unital C∗ -algebra, we mean that they are invertible in the unitization.

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By applying the index formula to St , we obtain a norm-continuous path of projections in M2 (B) ⊂ M2 (B(H )). For the sake of simplicity, let us assume that we have a norm-continuous path of projections (Tt ) in B itself. Choose A0 and A1 to be bases choices corresponding to (H, φ0 ) and (H, φ1 ) respectively, such that Ti ∈ Ck∗ (X, Ai ), i = 0, 1. Now we are in the position to apply the following Lemma 9.3, which finishes the proof for the even case. 2 Lemma 9.3. Let H be a Hilbert space, φ : C0 (X) → B(H ) a ∗ -representation. Denote B = Θ(φ) ⊂ B(H ), the C∗ -algebra generated by φ-uniform operators with φ-finite propagation. Assume that Tt , t ∈ [0, 1] is a homotopy of projections in B, and that A0 and A1 are two bases choices, such that Ti ∈ Ck∗ (X, Ai )10 . Then [T0 ] = [T1 ] ∈ K0 (Ck∗ (X, A0 , A1 )). Proof. Since Tt is a homotopy of projections in a C∗ -algebra B, there exists an invertible element v0 ∈ B with v0  = 1, such that T1 = v0−1 T0 v0 (see e.g. [2, Proposition 4.3.2]). Note that v0 might not have finite propagation, so we will need to make some approximations further on. The images of T0 and T1 under the inclusions of Ck∗ (X, A0 ) and Ck∗ (X, A1 ) into     Ck∗ (X, A0 , A1 ) ⊂ M2 (B(H )) are the operators T00 00 and 00 T01 . These two projections are    0 0 Murray–von Neumann equivalent by the elements x = 00 T00v0 and y = v −1 T0 0 . To finish the 0 argument, we must show that x, y ∈ Ck∗ (X, A0 , A1 ). For the rest of the proof, we will think of Mk = Mk (C) as B(span(e1 , . . . , ek )) in K (2 N), where e1 , e2 , . . . is the standard basis of 2 N. Let A ⊂ B(2 (Y ) ⊗ 2 N) be the algebra of all finite propagation matrices (tyz )y,z∈Y for which there exists k ∈ N with tyz ∈ Mk for all y, z ∈ Y . Then Ck∗ (Y ) is the norm closure of A. We shall give a proof that y ∈ Ck∗ (X, A0 , A1 ); a proof for x is analogous. Denote u0 = uA0 and u1 = uA1 . We need to show that y ∈ Ad(u0 ⊕ u1 )(M2 (Ck∗ (Y ))). This will follow from the following statement: For any ε > 0, there exists p ∈ A, such that p − u1 v0−1 T0 u∗0  < ε. By the choice of A0 and A1 , we know that there are sˆ0 , sˆ1 ∈ B(H ), such that u0 sˆ0 u∗0 , u1 sˆ1 u∗1 ∈ A, ˆs0 − T0  < ε and ˆs1 − v0−1 T0 v0  < ε. Note that sˆ0 and sˆ1 have finite propagation. Furthermore, there exists an invertible element v ∈ B with finite propagation, norm 1, and v − v0  < ε and v −1 − v0−1  < ε. It follows that v sˆ1 v −1 − T0  < 3ε. At this moment, the setting is as follows: we have finite propagation operators v, sˆ0 , sˆ1 and T0 , such that ˆs0 − T0  < ε, v sˆ1 v −1 − T0  < 3ε. Claim 4. There exists p ∈ A, such that p − u1 sˆ1 v −1 u∗0  < 4ε. Proof of claim. Combining the two inequalities with T0 gives       4ε > sˆ0 − v sˆ1 v −1  = v −1 sˆ0 − sˆ1 v −1   u1 v −1 u∗0 u0 sˆ0 u∗0 − u1 sˆ1 u∗1 u1 v −1 u∗0 . 10 For any bases choice A , C ∗ (X, A ) ⊂ B. The uniformity of T ∈ C ∗ (X, A ) follows the formula f =  f χ . Vy y k k Note that for fixed R  0 and f ∈ CR (X), there is a uniform bound on the number of nonzero terms in the sum by

bounded geometry.

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Denoting w = u1 v −1 u∗0 ∈ B(2 (Y ) ⊗ 2 (N)), s0 = u0 sˆ0 u∗0 ∈ A, s1 = u1 sˆ1 u∗1 ∈ A, we obtain ws0 − s1 w < 4ε. Note that w has finite propagation. Let k be such that all entries s0 and s1 belong to Mk . We split the standard basis of 2 (Y ) ⊗ 2 N into two sets B1 (first k vectors from each {y} ⊗2 N) With respect to this decomposition, we can  and B  other basis  w vectors).  2 (the w  write s0 = 0 00 , s1 = s011 00 and w = 11 12 . Consequently,    4ε > ws0 − s1 w =    Hence s w12  < 4ε. Denoting p =  0 s w 11   11 12  < 4ε. 2 0

0 0 s



 −

11 w11

0

s11 w11 0

     s11 w12   =   s11 w12  .    0  0

0 , we immediately see that p 0

∈ A and s1 w −p =

0

Returning to the proof of the lemma, we conclude     p − u1 v −1 T0 u∗   p − u1 v −1 T0 u∗  + εT0  0 0 0    p − u1 sˆ1 v −1 u∗0  + εT0  + 3ε < 4Cε + εT0  + 3ε. This finishes the proof.

2

Q+1 Proof of the 1-case. The operator ind(Q) = exp(−2πi Q+1 2 ) − 1 ∈ B(H ) is uniform (P = 2 2 satisfies P ∼ua P and so exp(−2πiP ) − 1 ∼ua P (exp(−2πi) − 1) = 0), but might not have finite propagation. However, from the formula for ind(Q) and finite propagation of Q it follows that ind(Q) − 1 ∈ Θ(φ), and so the existence of suitable A follows from Lemma 8.10 (after we have fixed some quasi-latticing partition (Vy )y∈Y ). We reduce the independence of the index on homotopies to independence on bases choices. Taking a homotopy (H, φt , Qt ) of 1-uniform Fredholm modules, we assume that all Qt have finite propagation. It follows that Ut = ind(Qt ), t ∈ [0, 1] is a homotopy of invertibles in B + = Θ(φ0 )+ . Since the set of invertibles is open, by a standard compactness argument we can assume that the homotopy is piecewise-linear. Hence, it is sufficient to assume that we have just one linear path of invertibles from (say) U0 to U1 in B + , and that we are given two bases choices A0 and A1 , such that Ui ∈ Ck∗ (X, Ai )+ , i = 1, 2. Applying Lemma 8.11 gives a bases choice A , such that Ai  A for each i = 1, 2. Hence U0 , U1 and the whole (linear) homotopy between them is actually in Ck∗ (X, A )+ . So [U0 ] = [U1 ] ∈ K1 (Ck∗ (X, A )), and the assertion will follow from the independence of the index on the choice of a bases choice. We find ourselves in the following situation: we are given an invertible U = 1 + K, K ∈ B = Θ(φ), and two bases choices A0 , A1 , such that K ∈ Ck∗ (X, Ai ), i = 0, 1. We will think of Mk = Mk (C) as B(span(e1 , . . . , ek )) in K (2 N), where e1 , e2 , . . . is the standard basis of 2 N. Let A ⊂ B(2 (Y ) ⊗ 2 N) be the algebra of all finite propagation matrices (tyz )y,z∈Y for which there exists k ∈ N with tyz ∈ Mk for all y, z ∈ Y . Then Ck∗ (Y ) is the norm closure of A. Denote u0 = uA0 and u1 = uA1 .     We will prove that U0 10 ∼ 10 U0 ∈ Ck∗ (X, A0 , A1 )+ . The standard rotation homotopy be sin2 ( π t) cos( π t) sin( π t) 2 2 2 K, and so it is suffitween these two matrices has the form 1 + cos( π2 t) sin( π2 t) cos2 ( π2 t)  0 K  0 0 cient to prove that actually 0 0 and K 0 ∈ Ck∗ (X, A0 , A1 ). Equivalently, that u0 Ku∗1 and u1 Ku∗0 ∈ A.

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Pick ε > 0. Since K ∈ Ck∗ (X, Ai ), i = 0, 1, there exist sˆ0 , sˆ1 ∈ B(H ) with finite propagation, such that si := ui sˆi u∗i ∈ A and ˆsi − K < ε for i = 0, 1. Since K ∈ B, there exist an operator ˆ < ε, i = 0, 1. Consequently, ˆsi − K ˆ < 2ε Kˆ ∈ B with finite propagation, such that K − K for i = 0, 1. At this moment, we can apply the proof of Claim 4 above (with v = 1 and T0 = K, otherwise verbatim), to obtain p ∈ A, such that p − u1 Ku∗0  < 8ε. Letting ε → 0, we obtain that u1 Ku∗0 ∈ A. Analogous proof shows also u0 Ku∗1 ∈ A. We are done. 2 10. On the Baum–Connes conjecture with ∞ -coefficients As an application of the Mayer–Vietoris sequence for uniform K-homology, we exhibit a connection with the Baum–Connes conjecture. Yu [14] proved that for a discrete group Γ , the Baum–Connes conjecture [1] for Γ with coefficients in ∞ (Γ, K ) is equivalent to the Coarse Baum–Connes conjecture for Γ . The righthand side in both conjectures is the K-theory of the Roe C∗ -algebra C ∗ Γ ∼ = ∞ (Γ, K ) r Γ . The core of Yu’s proof is showing the left-hand sides are the same, i.e. that      top  K∗ Γ, ∞ (Γ, K ) = lim KK∗Γ C0 ρ −1 () , ∞ (Γ, K ) ∼ = lim K∗ (Pd Γ ), ⊂BΓ,  compact

d→∞

where ρ : EΓ → BΓ denotes the quotient map. We prove an analogous statement for uniform K-homology in certain cases: Theorem 10.1. If Γ is a torsion-free countable discrete group, then     lim KK∗Γ C0 ρ −1 () , ∞ Γ ∼ = lim K∗u (Pd Γ ). ⊂BΓ,  compact

d→∞

As a consequence, this provides a computation of limd→∞ K∗u (Pd Γ ) for torsion-free discrete groups for which the Baum–Connes conjecture with commutative coefficients is known, for instance Zn or the free groups (see e.g. [7,10]). We consider a discrete group Γ endowed with a proper, left-invariant metric. Such a metric makes Γ into a uniformly discrete space with bounded geometry. There are such metrics on every discrete group, and any two such are quasi-isometric. For instance, if Γ is finitely generated, the word metric provides an example of such a metric. Proof of 10.1. First, realize that for countable discrete groups       lim KK∗Γ C0 ρ −1 () , ∞ Γ ∼ = lim KK∗Γ C0 (Pd Γ ), ∞ Γ . ⊂BΓ,  compact

d→∞

The rest of the proof is devoted to showing that the right-hand side above is in fact isomorphic to limd→∞ K∗u (Pd Γ ), by using the Mayer–Vietoris sequence. We proceed similarly as in [14, Proof of Theorem 2.7]. Let X be a Γ -invariant subset of ρ −1 (), where  ⊂ BΓ is compact. We construct a homomorphism   ψ : K∗u (X) → KK∗Γ C0 (X), ∞ Γ

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as follows: given a uniform Fredholm module (H, φ, F ) for X, we let H  = H ⊗ ∞ Γ ∼ = ∞ (Γ, H ), a Hilbert module over ∞ Γ . The group Γ acts on it by translations. Furthermore, we define φ  : C0 (X) → B(H  ) by      φ (f )ξ (γ ) = φ(γ ∗ f ) ξ(γ ) for f ∈ C0 (X), ξ ∈ H  ∼ = ∞ (Γ, H ), γ ∈ Γ , and where γ ∗ denotes the action of γ on C0 (X).  Finally, we put F ∈ B(H  ) to be the operator given by (F  ξ )(γ ) = F (ξ(γ )). It is straightforward to check that the triple (H  , φ  , F  ) is a Fredholm Γ -module. Since Γ acts on X by isometries, note that γ ∗ f has the same support and the same L-continuity as f , and so the size of matrices which approximate expressions like (F 2 − 1)φ(γ ∗ f ) does not depend on γ , only on f . We let the image of [(H, φ, F )] under ψ to be [(H  , φ  , F  )]. It is immediate that this assignment is well-defined, and it describes a group homomorphism. We prove that ψ is an isomorphism for X = Pd Γ . For X = Pd Γ , there exists a finite cover {Ui }m i=1 of X, such that each Ui is a space of the form Γ × Y , where Y is contractible, compact, with the diameter at most 12 . Such a cover can be constructed by considering sufficiently fine barycentric subdivision of the finite simplicial complex Pd Γ /Γ , and then pulling it back to Pd Γ by ρ. We now use the Mayer–Vietoris sequences for both KK∗Γ and K∗u (Theorem 5.1) simultaneously for showing that ψ is an isomorphism for X = Pd Γ . Note that ψ commutes with all the involved Mayer–Vietoris sequences. This, together with an induction process, reduces the general case to the case when X = Γ × Y , Y is as above. By [14, Lemma 2.3], KK∗Γ (C0 (Γ × Y ), ∞ Γ ) can be identified with KK∗ (C0 (Y ), ∞ Γ ). Under this identification, the map   ψ : K∗u (Γ × Y ) → KK∗ C0 (Y ), ∞ Γ can be understood as follows: Given a uniform Fredholm module (H, φ, F ) for Γ × Y , denote Hγ = φ(χ{γ }×Y )H . Then H  = γ ∈Γ Hγ is naturally a Hilbert module over ∞ Γ (with the coordinate-wise inner product). Furthermore, we let φ  : C0 (Y ) → B(H  ) be defined as ((φ  (f ))ξ )(γ ) = φ(χ{γ }×Y ·f )(ξ(γ )) for f ∈ C0 (Y ), ξ ∈ H  , γ ∈ Γ . Finally, define F  ∈ B(H  )  by F = γ ∈Γ φ(χ{γ }×Y )F φ(χ{γ }×Y ). With this notation, ψ assigns the Fredholm module [(H  , φ  , F  )] to [(H, φ, F )]. Now it is easy to see that ψ is an isomorphism, since we may assume that F has propagation at most 12 , by Proposition 7.4, Remark 7.2 and the properties of the space Γ × Y . This concludes the proof. 2 Remark 10.2. From the constructions in the above proof, the isomorphism commutes with the uniform index map. Moreover, from the definition of the index map it is clear that in fact it coincides with the Baum–Connes map with coefficients in ∞ Γ when Γ is torsion-free. Corollary 10.3. The statement that   μu : lim K∗u (Pd Γ ) → K∗ Cu∗ Γ d→∞

is an isomorphism for a torsion-free countable discrete group Γ (an analogue of the Coarse Baum–Connes conjecture) is equivalent to the Baum–Connes conjecture for Γ with coefficients in ∞ Γ .

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Remark 10.4. It is likely that the conclusion of Theorem 10.1 holds without any assumption on torsion. However, that would require at least some degree of homotopy invariance of the uniform K-homology, which would allow us to pass from EΓ to EΓ on the K-homology side (cf. [14, Lemma 2.10]). 11. Amenability As an application of uniform K-homology, we prove a criterion for amenability. It is analogous to similar criteria in the context of uniformly finite homology [3] and K-theory of uniform Roe algebras [6]. Both of these can be interpreted as saying that a space (which is uniformly discrete and has bounded geometry) is amenable if and only if its “fundamental class” is nontrivial in the appropriate group. In [3], it is the usual fundamental class in the 0th uniformly finite homology group; in [6] it is the class of the identity operator [1] in the K0 group of the uniform Roe algebra. Our criterion (Theorem 11.2) has the same form. Recall (Følner’s) definition of amenability (see [3, Section 3]). Definition 11.1. Let Y be a uniformly discrete metric space. For a set U ⊂ Y , we define its r-boundary by    ∂r U = y ∈ Y  d(y, U )  r and d(y, Y \ U )  r . We say that Y is amenable, if for any r, δ > 0, there exists a finite set U ⊂ Y , such that |∂r U | < δ. |U | Note that this definition is equivalent to the usual definition of amenability of groups (existence of an invariant mean) for spaces arising as Cayley graphs of discrete groups. However, we do not require the Følner sets to exhaust the whole space, and so we need to be cautious when applying this to general metric spaces. For instance, taking any uniformly discrete metric space Y , one can make it amenable by attaching an infinite “spaghetti” to it, i.e. an infinite ray. Also note that any “coarse disjoint union finite spaces” is also amenable in this sense, since for a given r > 0, we can always select a finite piece U of the space, which is at least r-far from the rest of the space, hence making ∂r U = ∅. In particular, this applies to expanders. Let X be a graph (with the edges attached) and let Y be its vertex set. Recall the definition of the fundamental class S ∈ K0u (X) (see Example 2.9). Let H = 2 Y ⊗ 2 N, and endow H with the multiplication action of C0 (X). Let S ∈ B(2 N) be the unilateral shift. Let S˜ = diag(S) ∈ B(H ) ˜ =1⊗ ˜ It is easy to see that S ∈ K u (X), and that ind(S) and finally denote S = [(H, φ, S)]. 0 2 2 2 p0 ∈ B( Y ⊗  N), where p0 is a rank one projection (onto Ce1 ∈  N). We also denote by 0 ∈ K0u (Y ) the trivial element. Theorem 11.2. Let X be a connected graph with the vertex set Y . Then Y is amenable if and only if S = 0 in K0u (X). More generally, if X is not connected, then Y is amenable if and only if there exists C  0, such that S = 0 in K0u (PC (Y )) (recall that PC (Y ) denotes the Rips complex of Y , see Definition 6.3). Remark 11.3. Note that the technical assumption that Y is a graph is not too restrictive, since every metric space with bounded geometry is coarsely equivalent to a graph.

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Proof. If Y is amenable, then μu (S) = [1] = [0] = μu (0) ∈ K0 (Cu∗ Y ) by [6], and so S = 0. For the convenience of the reader, let us sketch this part of Elek’s proof. The idea is that if Y is amenable, then usingFølner sets Bn , one can construct a trace on Cu∗ Y as an ultralimit of functions fn (T ) = |B1n | x∈Bn txx . Trace then distinguishes [1] from [0] in K0 (Cu∗ Y ). Let us turn to the reverse implication. Assume that Y is not amenable. We will proceed to constructing a homotopy connecting S and 0 in K0u (X).   First, we describe a “building block”. Denote I = [0, 1]. Denote T0 = 10 S0 and T1 =  S 0 ∈ B(HI ), where HI = 2 N ⊕ 2 N. Let the action ψ of C(I ) on HI be ψ(f )(η ⊕ ξ ) = 01 f (0)η ⊕ f (1)ξ . Let us show a homotopy (HI , ψt , Tt ) between (HI , ψ, T0 ) and (HI , ψ, T1 ). Define ⎧ 1 ⎪ ⎨ f (0)η ⊕ f (1 − 3t)ξ, 0  t  3 , 1 2 ψt (f )(η ⊕ ξ ) = f (0)η ⊕ f (0)ξ, 3  t  3, ⎪ ⎩ f (0)η ⊕ f (3t − 2)ξ, 23  1, and ⎧ ⎪ ⎨ T0,  Tt = αt 10 S0 αt∗ , ⎪ ⎩ T1 ,

0  t  13 , 1 3 2 3

 t  23 ,  1,

 cos( π (3t−1)) sin( π (3t−1))   k  where αt = − sin(2π (3t−1)) cos( π2 (3t−1)) is the rotation homotopy. It is clear that operators S 0l 0 S 2  S k−1 0  2 and (on the same Hilbert space with the same action of C(I )) are homotopic as well. 0 S l+1 Now we turn to Y ⊂ X. Assuming non-amenability of Y and applying [3, Theorem 3.1 and y Lemma 2.4], for each y ∈ Y there exists a “tail”, i.e. a sequence (zi )i0 ⊂ Y , such that z0 = y, y y C = supy,i (d(zi , zi+1 )) < ∞, satisfying the condition that in every ball of a fixed radius, the number of tails passing through is uniformly bounded. In the case when X is connected, we can reduce the general C to the case C = 1, i.e. to the situation when the tails actually follow the edges of X. We may achieve this just by refining the tails, without violating the condition on uniform bound on tails passing through balls, since Y has bounded geometry. If we do not assume connectedness, we may get by working with the Rips complex PC (Y ) instead of X = P1 (Y ), since any two points with distance  C are connected by an edge in PC (Y ). Consequently, it is possible to partition the collection of edges contained in all tails y y ((zi , zi+1 ))y∈Y,i∈N (we allow for multiplicities) into finitely many parts A1 , . . . , Ak , such that no two edges from the same part share a common vertex. The idea of the rest of the construction is to “send off” the S˜ along the tails off to infinity, and thus connecting S˜ with 1. This is done in k steps. In step j , we simultaneously apply the building block construction to each of the edges in Aj (this is possible by the choice of Aj ), thus “transferring” one S along each of those edges. After each step, we obtain a diagonal matrix in ˜ and ends B(H ) with various powers of S on the diagonal. The whole homotopy begins with S, with 1, since after all k steps the S from each y ∈ Y was shifted away from y along the tail. 2

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Acknowledgment The author would like to thank Guoliang Yu for helpful and enlightening conversations and never-ending encouragement. References [1] P. Baum, A. Connes, N. Higson, Classifying space for proper actions and K-theory of group C ∗ -algebras, in: C ∗ Algebras: 1943–1993, San Antonio, TX, 1993, in: Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240–291. [2] B. Blackadar, K-Theory for Operator Algebras, second ed., Math. Sci. Res. Inst. Publ., vol. 5, Cambridge Univ. Press, Cambridge, 1998. [3] J. Block, S. Weinberger, Aperiodic tilings, positive scalar curvature, and amenability of spaces, J. Amer. Math. Soc. 5 (4) (1992) 907–918. [4] J. Block, S. Weinberger, Large scale homology theories and geometry, in: Geometric Topology, Athens, GA, 1993, in: AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 522–569. [5] J. Brodzki, G.A. Niblo, N.J. Wright, Property A, partial translation structures, and uniform embeddings in groups, J. Lond. Math. Soc. (2) 76 (2) (2007) 479–497. [6] G. Elek, The K-theory of Gromov’s translation algebras and the amenability of discrete groups, Proc. Amer. Math. Soc. 125 (9) (1997) 2551–2553. [7] N. Higson, G.G. Kasparov, E-theory and KK-theory for groups which act properly and isometrically on Hilbert space, Invent. Math. 144 (1) (2001) 23–74. [8] N. Higson, J. Roe, Analytic K-Homology, Oxford Math. Monogr., Oxford Univ. Press, Oxford Sci. Publ., Oxford, 2000. [9] G.G. Kasparov, Equivariant KK-theory and the Novikov conjecture, Invent. Math. 91 (1) (1988) 147–201. [10] V. Lafforgue, K-théorie bivariante pour les algèbres de Banach et conjecture de Baum–Connes, Invent. Math. 149 (1) (2002) 1–95. [11] J. Roe, An index theorem on open manifolds. I, II, J. Differential Geom. 27 (1) (1988) 87–113, 115–136. [12] J. Roe, Index Theory, Coarse Geometry, and Topology of Manifolds, CBMS Reg. Conf. Ser. Math. 90, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. [13] J. Roe, Lectures on Coarse Geometry, Univ. Lecture Ser. 31, Amer. Math. Soc., Providence, RI, 2003. [14] G. Yu, Baum–Connes conjecture and coarse geometry, K-Theory 9 (3) (1995) 223–231. [15] G. Yu, The coarse Baum–Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math. 139 (1) (2000) 201–240.

Journal of Functional Analysis 257 (2009) 122–148 www.elsevier.com/locate/jfa

Stable manifolds for nonuniform polynomial dichotomies ✩ António J.G. Bento ∗ , César Silva Departamento de Matemática, Universidade da Beira Interior, 6201-001 Covilhã, Portugal Received 17 October 2008; accepted 29 January 2009 Available online 24 February 2009 Communicated by L. Gross

Abstract We establish the existence of smooth stable manifolds in Banach spaces for sufficiently small perturbations of a new type of dichotomy that we call nonuniform polynomial dichotomy. This new dichotomy is more restrictive in the “nonuniform part” but allow the “uniform part” to obey a polynomial law instead of an exponential (more restrictive) law. We consider two families of perturbations. For one of the families we obtain local Lipschitz stable manifolds and for the other family, assuming more restrictive conditions on the perturbations and its derivatives, we obtain C 1 global stable manifolds. Finally we present an example of a family of nonuniform polynomial dichotomies and apply our results to obtain stable manifolds for some perturbations of this family. © 2009 Elsevier Inc. All rights reserved. Keywords: Invariant manifolds; Nonautonomous dynamics; Nonuniform polynomial dichotomies

1. Introduction The existence of invariant manifolds is one of the key features in the theory of nonuniform hyperbolicity. The concept of nonuniform hyperbolicity introduced by Pesin [20–22] (see [1,2] for a description of the current status of the theory) is a generalization of the classical concept of (uniform) hyperbolicity. In the nonuniform hyperbolic context the rates of expansion and ✩

Supported by Centro de Matemática da Universidade da Beira Interior.

* Corresponding author.

E-mail addresses: [email protected] (A.J.G. Bento), [email protected] (C. Silva). URL: http://www.mat.ubi.pt/~csilva (C. Silva). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.01.032

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123

contraction are allowed to vary from point to point. The stable manifold theorem for nonuniform hyperbolic trajectories obtained by Pesin [20] in the finite-dimensional setting is an elaboration of the classical work of Perron. Since then other proofs were obtained, namely, in [24] Ruelle gave a proof based on the study of perturbations of products of matrices occurring in Oseledets’ multiplicative ergodic theorem [18]. The proof given by Pugh and Shub in [23] is based on the classical work of Hadamard and uses graph transform techniques. There exist also versions of the stable manifold theorem for dynamical systems in infinite-dimensional spaces. In [25] Ruelle established a corresponding version in Hilbert spaces under some compactness assumptions, following his approach in [24]. In [17] Mañé considered transformations in Banach spaces under some compactness and invertibility assumptions that includes the case of differentiable maps with compact derivative at each point. The results of Mañé were generalized by Thieullen in [27] for a family of transformations satisfying some asymptotic compactness. The existence of invariant manifolds is also an important subject in the context of exponential dichotomies introduced by Perron in [19]. There is a substantial amount of literature concerning the existence of stable and unstable manifolds for exponential dichotomies (see for example [26] and the references given there). As mentioned before, the notion of nonuniform hyperbolicity is a generalization of the concept of (uniform) hyperbolicity. Similarly there is also a concept introduced by Barreira and Valls [9,6] of nonuniform exponential dichotomy that is a weaker (and therefore more general) version of the classical notion of exponential dichotomy. In the discrete time setting, Barreira and Valls obtained C 1 stable manifolds for nonuniformly exponential dichotomies in finite dimension in [7]. Building on this result Barreira, Silva and Valls were able in [3] to establish the existence of C k local manifolds for C k perturbations, using an induction process and considering a more geometric approach based on the linear extension of the dynamics. Assuming some exponential decay of the derivatives along the orbits, the same authors established in [4] the existence of C 1 global manifolds for perturbations of nonuniformly exponential dichotomies in Banach spaces. Continuous time versions of this results were obtained by Barreira and Valls in [5,6,8,11]. In a recent work Barreira and Valls [12] considered a generalization of the concept of nonuniform exponential dichotomies that they call ρ-nonuniform exponential dichotomy, with ρ an + increasing differentiable function from R+ 0 into R0 such that lim

t→+∞

log t = 0. ρ(t)

(1)

With this generalization, Barreira and Valls replaced the asymptotic rates ect that appear in the nonuniform exponential case by the asymptotic rates ecρ(t) . Barreira and Valls established, in a finite-dimensional space, the existence of ρ-nonuniform exponential dichotomies and trichotomies for a general family of nonautonomous linear differential equations v  = A(t)v, where A(t) are matrices in some block form. To achieve this Barreira and Valls used an adapted type of Lyapunov exponent. In this work we consider a different kind of nonuniform dichotomy where the rates of expansion and contraction are allowed to vary polynomially. Naturally, the nonuniform parts must vary at most polynomially (see (2) and (3)). We thus consider a new type of behavior: there are families of nonuniform polynomial dichotomies that are not nonuniform exponential dichotomies and vice versa. Even in the more general case of ρ-nonuniform exponential dichotomy, it is not possible to have a polynomial behavior because, to have a polynomial behavior, ρ(t) would have

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to increase at most logarithmically, and by (1) this is not possible. Note that our case allows situations for which the Lyapunov exponent defined in [10, Section 8] for Hilbert spaces is zero for all v ∈ E1 (see Section 2 for the definition of E1 ). The main results of this paper are stable manifolds theorems for perturbations of nonuniform polynomial dichotomies. In Section 3 we get local Lipschitz stable manifolds and in Section 4 we get global C 1 stable manifolds. The reason for the difference in the regularity of the manifolds obtained is that to get C 1 manifolds in the local case we would have to consider perturbations that are zero outside a ball of increasingly small radius and it is not known in the infinite-dimensional setting how to obtain appropriate cutoff functions (see the comment in [11, p. 2]). The content of the paper is the following: in Section 2 we introduce some notations, the main definitions and also establish a technical lemma used several times; then, respectively in Sections 3 and 4, we obtain a local and a global stable manifold theorem; finally we present in Section 5 examples of families of nonuniform polynomial dichotomies and apply our results to obtain stable manifolds for some perturbations of these families. 2. Notation and preliminaries Let B(X) be the space of bounded linear operators in the Banach space X. Given a sequence (An )n∈N of invertible operators of B(X) we define A(m, n) =



Am−1 . . . An Id

if m > n, if m = n.

We say that (An )n∈N admits a nonuniform polynomial dichotomy if there exist projections Pn : X → X for n ∈ N such that Pm A(m, n) = A(m, n)Pn ,

m, n ∈ N,

and constants a < 0  b, ε  0 and D  1 such that for every m  n,   A(m, n)Pn   D(m − n + 1)a nε ,   A(m, n)−1 Qm   D(m − n + 1)−b mε ,

(2) (3)

where Qm = Id −Pm is the complementary projection. When ε = 0 we say that we have a uniform polynomial dichotomy or simply a polynomial dichotomy. In these conditions we define, for each n ∈ N, the linear subspaces En = Pn (X) and Fn = Qn (X). Without loss of generality, we always identify the spaces En × Fn and En ⊕ Fn as the same space and we equip these spaces with the norm given by (x, y) = x + y for (x, y) ∈ En × Fn . We are going to address the problem of existence of stable manifolds of the dynamics given by  F(m, n) =

(Am−1 + fm−1 ) ◦ · · · ◦ (An + fn ) Id

if m > n, if m = n,

(4)

where (An )n∈N admits a nonuniform polynomial dichotomy and fn : X → X are perturbations that verify some conditions to be specified later.

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125

Given n ∈ N and vn = (ξ, η) ∈ En × Fn , for each m > n we write vm = F(m, n)(vn ) = (xm , ym ) ∈ Em × Fm . Writing fm = (gm , hm ) where gm = Pm+1 fm and hm = Qm+1 fm for each m  n, the trajectory (vm )mn satisfies the following equations xm = A(m, n)ξ +

m−1 

A(m, k + 1)gk (xk , yk ),

(5)

A(m, k + 1)hk (xk , yk ).

(6)

k=n

ym = A(m, n)η +

m−1  k=n

In what follows we are going  toαuse Dirichlet series. For every α < −1, we denote by λα the sum of the Dirichlet series ∞ k=1 k . The following lemma will be used several times. Lemma 1. Let m, n ∈ N with m > n, a < 0, q > 0 and ε  0. (a) If aq + ε < −1, then the following inequality holds m−1 

(m − k)a (k + 1)ε (k − n + 1)aq+a  2ε−a λaq+ε (m − n + 1)a nε .

(7)

k=n

(b) If ε > 0, then the following inequality holds m−1 

(m − k)a (k − n + 1)a (k + 1)ε k −3ε−1  2ε−a λ−2ε−1 (m − n + 1)a .

(8)

k=n

Proof. (a) Because the sum of the factors of (m − k)(k − n + 1) is constant and a < 0, it follows that (m − k)a (k − n + 1)a  (m − n)a  2−a (m − n + 1)a , k = n, . . . , m − 1. On the other hand, we have  ε n ε (k − n + 1)ε  2ε nε (k − n + 1)ε , (k + 1) = 1 + k−n+1 k = n, . . . , m − 1. Therefore m−1 

(m − k)a (k + 1)ε (k − n + 1)aq+a  2ε−a (m − n + 1)a nε

k=n

m−1 

(k − n + 1)aq+ε

k=n

2

ε−a

λaq+ε (m − n + 1)a nε .

(b) It follows immediately from (9) and (k + 1)ε  2ε k ε .

2

(9)

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3. Local stable manifolds In this section we assume that there are constants c > 0 and q > 1 such that the functions fn in the perturbed dynamics (4) verify the following conditions fn (0) = 0,  

fn (u) − fn (v)  cu − v u + v q

(10) (11)

for every n ∈ N and u, v ∈ X. Making v = 0 in (11) we have   fn (u)  cuq+1

(12)

for every u ∈ X. We denote by Bn (r) the open ball of En centered at zero and with radius r > 0. The initial condition at time n will be taken in Bn (δn−β ) for some δ, β > 0. We denote by Xβ the space of sequences (ϕn )n∈N of continuous functions ϕn : Bn (δn−β ) → Fn such that ϕn (0) = 0,   ϕn (ξ ) − ϕn (ξ¯ )  ξ − ξ¯ 

(13) (14)

for every ξ , ξ¯ ∈ Bn (δn−β ). For each ϕ = (ϕn )n∈N ∈ Xβ we define  −β

ϕn (ξ ) : n ∈ N and ξ ∈ Bn δn \ {0} . ϕ = sup ξ  



Clearly ϕ  1, and given m ∈ N and ξ = 0, we have   ϕn (ξ )  δn−β ϕn (ξ )  δϕ  δ ξ  for every ϕ ∈ Xβ . This readily implies that Xβ is a complete metric space with the distance induced by  ·  . We also consider the space X∗β of sequences ϕ = (ϕn )n∈N with ϕn : En → Fn such that the sequence (ϕn |Bn (δn−β ))n∈N is in Xβ and, for each n ∈ N,  ϕn (ξ ) = ϕn

δn−β ξ ξ 





whenever ξ ∈ / Bn δn−β .

There is a one-to-one correspondence between the sequences in Xβ and in X∗β because for each sequence of functions ϕ = (ϕn )n∈N ∈ Xβ there is a unique extension ϕ = (ϕ n )n∈N such that each ϕ n is a Lipschitz extension of ϕn to Bn (δn−β ). Clearly X∗β is also a complete metric space with the metric induced by X∗β  ϕ → ϕ|Xβ . Furthermore, one can easily verify that given ϕ ∈ X∗β and n ∈ N we have   ϕn (x) − ϕn (y)  2x − y for every x, y ∈ E.

(15)

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127

Given β, δ > 0 and (ϕn )n∈N ∈ Xβ , for each n ∈ N, we consider the graph Vn,δ,β =





ξ, ϕn (ξ ) : ξ ∈ Bn δn−β .

(16)

We now present the main result of this section. Theorem 1 (Local stable manifolds). Let (An )n∈N be a sequence of invertible bounded linear operators, acting on a Banach space X, that admits a nonuniform polynomial dichotomy satisfying (2) and (3) for some D  1, a < 0  b and ε > 0 and let (fn )n∈N be a sequence of functions, acting on X, that verifies (10) and (11) for some c > 0 and some q > 1. If aq + ε + 1 < 0 and a + β < 0 hold with β = ε(1 + 2/q), then, for every C > D, choosing δ sufficiently small, there is a unique ϕ ∈ Xβ such that F(m, n)(Vn, δ ,β+ε ) ⊂ Vm,δ,β C

for every m  n,

(17)

with F(m, n) given by (4) and Vn, δ ,β+ε and Vm,δ,β given by (16). C Furthermore, for every m  n and ξ , ξ¯ ∈ Bn (δn−(β+ε) /C) we have 



 F(m, n) ξ, ϕn (ξ ) − F(m, n) ξ¯ , ϕn (ξ¯ )   2C(m − n + 1)a nε ξ − ξ¯ .

(18)

We call each Vn,δ,β a stable manifold. In view of the forward invariance in (17), each trajectory starting Vn, δ ,β+ε must be in Vm,δ,β . C Thus, for every (ξ, ϕn (ξ )) ∈ Vn, δ ,β+ε , using Eqs. (5) and (6), we have to prove that C

xm (ξ ) = A(m, n)ξ +

m−1 



A(m, k + 1)gk xk (ξ ), ϕk xk (ξ ) ,

(19)

k=n m−1 



A(m, k + 1)hk xk (ξ ), ϕk xk (ξ ) , ϕm xm (ξ ) = A(m, n)ϕn (ξ ) +

(20)

k=n

and   xm (ξ )  δm−β

(21)

for every ξ ∈ Bn (δn−(β+ε) /C) and every m > n, where



F(m, n) ξ, ϕn (ξ ) = xm (ξ ), ϕm xm (ξ ) ∈ Em × Fm . The idea of the proof of Theorem 1 is to solve Eqs. (19) and (20) separately using in each case the Banach fixed point theorem. For this we first establish, using the Banach fixed point theorem on a suitable space B, that for every ϕ ∈ Xβ there is a unique sequence of functions x ϕ = (xm )mn ∈ B that verifies (19) and (21). To prove Eq. (20), we first prove that this equation

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is equivalent to another one and we solve this second equation applying the Banach fixed point theorem in the space X∗β . Let B = Bn,β be the space of all sequences x = (xm )mn of functions

xm : Bn δn−β → Em such that for every m  n and every ξ, ξ¯ ∈ Bn (δn−β ) we have xn (ξ ) = ξ, xm (0) = 0,   xm (ξ ) − xm (ξ¯ )  C(m − n + 1)a nε ξ − ξ¯ , for some constant C > D. Making ξ¯ = 0 in (23) we obtain the following estimate   xm (ξ )  C(m − n + 1)a nε ξ   Cδ(m − n + 1)a nε−β

(22) (23)

(24)

for every m  n and every ξ ∈ Bn (δn−β ). This space B allows to estimate the speed of decay of the stable component of the solution along the graphs given by ϕ. In fact, if ξ ∈ Bn (δn−(β+ε) /C), then, (21) holds because a + β < 0:   xm (ξ )  C(m − n + 1)a nε ξ   δ(m − n + 1)a n−β  δm−β . For every x ∈ B, we define  x = sup 

 −β

xm (ξ ) : m  n, ξ ∈ Bn δn (m − n + 1)a nε

(25)

and with the metric induced by (25), B is a complete metric space. Given ϕ = (ϕn )n∈N ∈ X∗β and x = (xm )mn ∈ B we write ∗ ϕm = ϕm ◦ xm



∗ and fm∗ (ξ ) = fm xm (ξ ), ϕm (ξ ) .

Lemma 2. Given δ > 0 sufficiently small, for each ϕ ∈ X∗β and n ∈ N there exists a unique sequence x = x ϕ ∈ B satisfying Eq. (19) for every m  n and ξ ∈ Bn (δn−β ). Proof. We define an operator J in B by (J x)n (ξ ) = ξ , and for each m > n by (J x)m (ξ ) = A(m, n)ξ +

m−1 



A(m, k + 1)gk xk (ξ ), ϕk xk (ξ ) .

k=n

One can easily verify from (22), (13) and (10) that (J x)m (0) = 0 for every m  n. From the definition of the operator it follows that   (J x)m (ξ ) − (J x)m (ξ¯ )       m−1 A(m, k + 1)Pk+1  · f ∗ (ξ ) − f ∗ (ξ¯ ).  A(m, n)Pn ξ − A(m, n)Pn ξ¯  + k k k=n

(26)

A.J.G. Bento, C. Silva / Journal of Functional Analysis 257 (2009) 122–148

129

Using (2) we have     A(m, n)Pn ξ − A(m, n)Pn ξ¯   A(m, n)Pn ξ − ξ¯   D(m − n + 1)a nε ξ − ξ¯  and from (2), (11), (15), (23) and (7) we also have m−1 

    A(m, k + 1)Pk+1  · f ∗ (ξ ) − f ∗ (ξ¯ ) k k

k=n

 cD

m−1 

  



(m − k)a (k + 1)ε xk (ξ ) − xk (ξ¯ ) + ϕk xk (ξ ) − ϕk xk (ξ¯ ) 

k=n

    

 

 q × xk (ξ ) + ϕk xk (ξ )  + xk (ξ¯ ) + ϕk xk (ξ¯ )   cD

m−1 

     q  (m − k)a (k + 1)ε 3xk (ξ ) − xk (ξ¯ ) 3xk (ξ ) + 3xk (ξ¯ )

k=n

 c2q (3C)q+1 Dδ q nε(q+1)−βq ξ − ξ¯ 

m−1 

(m − k)a (k + 1)ε (k − n + 1)aq+a

k=n

2

q+ε−a

q+1

c(3C)

Dδ λaq+ε (m − n + 1)a nε(q+2)−βq ξ − ξ¯ . q

Choosing δ sufficiently small, it follows from (26) that   (J x)m (ξ ) − (J x)m (ξ¯ )  C(m − n + 1)a nε ξ − ξ¯  and this implies the inclusion J (B) ⊂ B. We now show that J is a contraction for the metric induced by (25). Let x, y ∈ B. Then    m−1  (J x)m (ξ ) − (Jy)m (ξ )  A(m, k + 1)Pk+1 αk , k=n

where αk = fk (xk (ξ ), ϕk (xk (ξ ))) − fk (yk (ξ ), ϕk (yk (ξ ))). By (11) and (15) we have   



αk  c xk (ξ ) − yk (ξ ) + ϕk xk (ξ ) − ϕk yk (ξ )      

 

 q × xk (ξ ) + ϕk xk (ξ )  + yk (ξ ) + ϕk yk (ξ )        q  3q+1 cxk (ξ ) − yk (ξ ) xk (ξ ) + yk (ξ ) and, using (24), it follows that   αk  2q 3q+1 cC q δ q (k − n + 1)aq n(ε−β)q xk (ξ ) − yk (ξ )  2q 3q+1 cC q δ q (k − n + 1)aq+a nε(q+1)−βq x − y . Hence, from (2) and (27), we have

(27)

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  (J x)m (ξ ) − (Jy)m (ξ ) 2 3

q q+1

q

q ε(q+1)−βq

cC Dδ n

x − y



m−1 

(m − k)a (k + 1)ε (k − n + 1)aq+a

k=n

2

q−a+ε q+1

3

cC Dδ λaq+ε (m − n + 1)a nε(q+2)−βq x − y q

q

and choosing δ sufficiently small it follows that   (J x)m (ξ ) − (Jy)m (ξ )  μ(m − n + 1)a nε x − y with μ < 1. Therefore, J x − Jy  μx − y , and J is a contraction in B provided that δ is sufficiently small. Because B is complete, by the Banach fixed point theorem, the map J has a unique fixed point x ϕ in B, which is thus the desired sequence. 2 ϕ

We now represent by (xn,k )kn ∈ Bn,β the unique sequence given by Lemma 2. Lemma 3. Given δ > 0 sufficiently small and ϕ ∈ X∗β the following properties hold: (1) If for every n ∈ N, m  n and ξ ∈ Bn (δn−β ) the identity (20) holds, then ϕn (ξ ) = −

∞ 

ϕ ϕ

A(k + 1, n)−1 hk xn,k (ξ ), ϕk xn,k (ξ )

(28)

k=n

for every n ∈ N, m  n and ξ ∈ Bn (δn−β ). (2) If for every n ∈ N, m  n and ξ ∈ Bn (δn−β ) Eq. (28) holds, then (20) holds for every ξ ∈ Bn (δn−(β+ε) /C). Proof. First we prove that the series in (28) is convergent. From (3), (12), (15) and (24), we conclude that ∞  

 A(k + 1, n)−1 hk x ϕ (ξ ), ϕk x ϕ (ξ )  n,k n,k k=n ∞   

 A(k + 1, n)−1 Qk+1 fk x ϕ (ξ ), ϕk x ϕ (ξ )   n,k n,k k=n



∞ 

  

 q+1 D(k − n + 2)−b (k + 1)ε c xn,k (ξ ) + ϕk xn,k (ξ ) 

k=n

 cD

∞ 

q+1 (k − n + 2)−b (k + 1)ε 3Cδ(k − n + 1)a nε−β k=n

A.J.G. Bento, C. Silva / Journal of Functional Analysis 257 (2009) 122–148

 c(3Cδ)q+1 Dn(ε−β)(q+1)

131

∞  (k − n + 1)aq+a−b (k + 1)ε k=n

 2ε c(3Cδ)q+1 Dnε(q+2)−β(q+1)

∞  (k − n + 1)aq+a−b+ε < ∞. k=n

Let us suppose that (20) holds. Then by A(m, n)−1 A(m, k + 1) = A(k + 1, n)−1 , Eq. (20) can be written in the following equivalent form  ϕ ϕ ϕ

m−1

(ξ ) − A(k + 1, n)−1 hk xn,k (ξ ), ϕk xn,k (ξ ) . ϕn (ξ ) = A(m, n)−1 ϕm xn,m

(29)

k=n

Using (3), (15) and (24), we have 

 

 A(m, n)−1 ϕm x ϕ (ξ )  = A(m, n)−1 Qm ϕm x ϕ (ξ )  n,m n,m   ϕ  2D(m − n + 1)−b mε xn,m (ξ )  2D(m − n + 1)−b mε Cδ(m − n + 1)a nε−β  2CDδ(m − n + 1)a−b+ε n2ε−β and this converges to zero when m → ∞. Hence, letting m → ∞ in (29) we obtain the identity (28). We now assume that for every n ∈ N, m  n and ξ ∈ Bn (δn−β ) the identity (28) holds. If ξ ∈ Bn (δn−(β+ε) /C) then, since a + β < 0, we get   xn,m (ξ )  C(m − n + 1)a nε ξ   δ(m − n + 1)a n−β  δm−β . (30) Then A(m, n)ϕn (ξ ) = −

∞ 

ϕ ϕ

A(m, k + 1)hk xn,k (ξ ), ϕk xn,k (ξ ) ,

k=n

and thus it follows from (28), the uniqueness of the sequences x ϕ and (30) that A(m, n)ϕn (ξ ) +

m−1 

ϕ ϕ

A(m, k + 1)hk xn,k (ξ ), ϕk xn,k (ξ )

k=n

=−

∞ 

ϕ ϕ

A(m, k + 1)hk xn,k (ξ ), ϕk xn,k (ξ )

k=m

=−

∞  k=m

ϕ ϕ

ϕ ϕ



A(m, k + 1)hk xm,k xn,m (ξ ) , ϕk xm,k xn,m (ξ )

ϕ

= ϕm xn,m (ξ ) . This proves the lemma.

2

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Lemma 4. Given δ > 0 sufficiently small, for each ϕ, ψ ∈ X∗β , n ∈ N, m  n and ξ ∈ Bn (δn−β ) we have   ϕ x (ξ ) − x ψ (ξ )  C (m − n + 1)a ξ ϕ − ψ . m m 2

(31)

 ϕ ϕ

ψ ψ

 γk = fk xk (ξ ), ϕk xk (ξ ) − fk xk (ξ ), ϕk xk (ξ ) ,

(32)

Proof. Putting

by (11) it follows that   ϕ

ψ 

 ϕ ψ γk  c xk (ξ ) − xk (ξ ) + ϕk xk (ξ ) − ψk xk (ξ )   ϕ   ϕ   ψ   ψ  q × xk (ξ ) + ϕk xk (ξ )  + xk (ξ ) + ψk xk (ξ )   ψ  q   ϕ

ψ   ϕ   ϕ ψ  c xk (ξ ) − xk (ξ ) + ϕk xk (ξ ) − ψk xk (ξ )  3xk (ξ ) + 3xk (ξ )   ϕ

 ϕ ψ 

ψ  c(6Cδ)q xk (ξ ) − xk (ξ ) + ϕk xk (ξ ) − ψk xk (ξ )  (k − n + 1)aq n(ε−β)q and because  ϕ



 

 

 ϕk x (ξ ) − ψk x ψ (ξ )   ϕk x ϕ (ξ ) − ϕk x ψ (ξ )  + ϕk x ψ (ξ ) − ψk x ψ (ξ )  k k k k k k  ϕ   ψ

ψ  ψ  xk (ξ ) − xk (ξ ) + ϕk xk (ξ ) − ψk xk (ξ )   ϕ   ψ  ψ  xk (ξ ) − xk (ξ ) + ϕ − ψ xk (ξ ) we have   ψ 

 ϕ ψ γk  c(6Cδ)q 2xk (ξ ) − xk (ξ ) + ϕ − ψ xk (ξ ) (k − n + 1)aq n(ε−β)q .

(33)

Using (33), we are going to prove this lemma by induction on m. For m = n the result follows ψ ϕ immediately because xn (ξ ) = ξ = xn (ξ ). Suppose that (31) is true for n, . . . , m − 1. Then for k = n, . . . , m − 1 we have  ϕ   ψ  ψ 2xk (ξ ) − xk (ξ ) + ϕ − ψ xk (ξ )  C(k − n + 1)a ξ  ϕ − ψ + C(k − n + 1)a nε ξ ϕ − ψ  2C(k − n + 1)a nε ξ ϕ − ψ and this implies by (33) that γk  2cC q+1 (6δ)q ξ ϕ − ψ (k − n + 1)aq+a nε(q+1)−βq . Hence we have

A.J.G. Bento, C. Silva / Journal of Functional Analysis 257 (2009) 122–148

133

  ϕ x (ξ ) − x ψ (ξ ) m



m

m−1 

  A(m, k + 1)Pk+1 γk

k=n

 2cC q+1 D(6δ)q nε(q+1)−βq ξ ϕ − ψ

m−1 

(m − k)a (k + 1)ε (k − n + 1)aq+a

k=n

2

1+ε−a

cC

q+1

D(6δ) λaq+ε (m − n + 1) n q

a ε(q+2)−βq

ϕ − ψ ξ .

Choosing δ > 0 sufficiently small and using the fact that ε(q + 2) − βq = 0 we obtain (31).

2

Lemma 5. Given δ > 0 sufficiently small there is a unique ϕ ∈ X∗β such that ϕn (ξ ) = −

∞ 

ϕ ϕ

A(k + 1, n)−1 hk xk (ξ ), ϕk xk (ξ )

k=n

for every n ∈ N and every ξ ∈ Bn (δn−β ). Proof. We consider the operator Φ defined for each ϕ ∈ X∗β by (Φϕ)n (ξ ) = −

∞ 

ϕ ϕ

A(k + 1, n)−1 hk xk (ξ ), ϕk xk (ξ ) ,

(34)

k=n ϕ

where x ϕ = (xk )kn is the unique sequence given by Lemma 2. Since x ϕ ∈ Bβ we have ϕ xm (0) = 0, m  n. It follows from (10), (22), (13) and (34) that (Φϕ)n (0) = 0 for each n ∈ N. Furthermore, given ξ, ξ¯ ∈ Bn (δn−β ), by (3) and (11), we have   (Φϕ)n (ξ ) − (Φϕ)n (ξ¯ ) 

∞      A(k + 1, n)−1 Qk+1  · f ∗ (ξ ) − f ∗ (ξ¯ ) k

k

k=n

 cD

∞       q  (k − n + 2)−b (k + 1)ε 3xk (ξ ) − xk (ξ¯ ) 3xk (ξ ) + 3xk (ξ¯ ) k=n

 3cC q+1 D(6δ)q nε(q+1)−βq ξ − ξ¯ 

∞  (k − n + 2)−b (k + 1)ε (k − n + 1)aq+a k=n

 2q+ε c(3C)q+1 Dδ q nε(q+2)−βq ξ − ξ¯ 

∞ 

(k − n + 1)aq+a−b+ε

k=n

 2 3cC ε

q+1

D(6δ) λaq+a−b+ε ξ − ξ¯ . q

Hence, choosing δ > 0 sufficiently small (independently of ϕ, n and ξ ) we have (14). Therefore Φ(X∗β ) ⊂ X∗β .

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We now show that Φ is a contraction. Given ϕ, ψ ∈ X∗β and n ∈ N, let x ϕ and x ψ be the unique sequences given by Lemma 2 respectively for ϕ and ψ . By (33) (see (32) for the definition of γk ) and Lemma 4 we have γk  c2C q+1 (6δ)q ξ ϕ − ψ (k − n + 1)aq+a nε(q+1)−βq and this inequality and (3) imply   (Φϕ)n (ξ ) − (Φψ)n (ξ ) 

∞    A(k + 1, n)−1 Qk+1 γk k=n

 2cC q+1 D(6δ)q ξ ϕ − ψ nε(q+1)−βq

∞ 

(k − n + 2)−b (k + 1)ε (k − n + 1)aq+a

k=n

 21+ε cC q+1 D(6δ)q ξ ϕ − ψ nε(q+2)−βq

∞  (k − n + 1)aq+a−b+ε k=n

2

1+ε

cC

q+1



D(6δ) ξ λaq+a−b+ε ϕ − ψ . q

Hence, choosing δ > 0 sufficiently small, we have   (Φϕ)n (ξ ) − (Φψ)n (ξ )  μξ ϕ − ψ for some μ < 1. This implies that Φϕ − Φψ  μϕ − ψ and this means that Φ is a contraction in X∗β . Therefore the map Φ has a unique fixed point ϕ in X∗β that is the desired sequence. 2 We are now in conditions to prove Theorem 1. Proof of Theorem 1. By Lemma 2, for each ϕ ∈ X∗β there is a unique sequence x ϕ ∈ B satisfying (19). It remains to solve (20) with x = x ϕ . By Lemma 3, this is equivalent to solve (28). Finally, by Lemma 5, there is a unique solution of (28). This establishes the existence of the stable manifolds for δ > 0 sufficiently small. Moreover, for each n ∈ N, m  n and ξ, ξ¯ ∈ Bn (δn−(β+ε) /C) it follows from (14) that    



  F(m, n) ξ, ϕn (ξ ) − F(m, n) ξ¯ , ϕn (ξ¯ )   xm (ξ ) − xm (ξ¯ ) + ϕ ∗ (ξ ) − ϕ ∗ (ξ¯ ) m m    2xm (ξ ) − xm (ξ¯ )  2C(m − n + 1)a nε ξ − ξ¯ . Hence we obtain (18) and the theorem is proved.

2

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135

4. Global stable manifolds We say that a function is of class C 1,1 if it is of class C 1 and its derivative is Lipschitz. We will consider that the perturbations fn in (4) are of class C 1,1 and we assume that there exists δ > 0 such that, for every n ∈ N and u, v ∈ X, fn (0) = 0,

d0 fn = 0,

du fn   δn−3ε−1 , du fn − dv fn   δn

−3ε−1

u − v,

(35) (36) (37)

with the same ε as in (2) and (3). In this section, for technical reasons, we have to assume that ε > 0. It is easy to verify that all the results in this section remain true in the case ε = 0 if we replace the exponent in (36) and (37) by −1 − γ with γ > 0. Let X be the space of sequences of C 1,1 functions ϕn : En → Fn such that, for every n ∈ N and x, y ∈ En , ϕn (0) = 0,

d0 ϕn = 0,

(38)

dx ϕn   1,

(39)

dx ϕn − dy ϕn   x − y.

(40)

Given (ϕn )n∈N ∈ X we consider the graphs



Vn = ξ, ϕn (ξ ) : ξ ∈ En

(41)

that we call global stable manifolds. We have the following result. Theorem 2 (Global stable manifolds). Let (An )n∈N be a sequence of bounded invertible linear operators, acting on a Banach space X, that admits a nonuniform polynomial dichotomy satisfying (2) and (3) for some D  1, a < 0  b and ε > 0 and let (fn )n∈N be a sequence of functions, acting on X, that verify (35), (36) and (37) for some δ > 0. If a + ε < b,

(42)

then, choosing δ > 0 sufficiently small, there exists a unique sequence (ϕn )n∈N ∈ X such that (a) F(m, n)(Vn ) = Vm for every m  n, where F(m, n) is given by (4) and Vn and Vm are given by (41); (b) Vn is a C 1,1 manifold with T0 Vn = En for each n ∈ N; (c) there exists K > 0 such that for m  n and ξ, ξ¯ ∈ En we have   F(m, n)(v) − F(m, n)(v) ¯   K(m − n + 1)a nε ξ − ξ¯ ,   dv F(m, n) − dv¯ F(m, n)  K(m − n + 1)a n2ε ξ − ξ¯ , where v = (ξ, ϕn (ξ )) and v¯ = (ξ¯ , ϕn (ξ¯ )).

(43) (44)

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The idea of the proof of this theorem follows the same lines of the proof of Theorem 1, although with different spaces of sequences of functions. As in Theorem 1, we have to solve the following equations

xm (ξ ) = A(m, n)ξ +

m−1 



A(m, k + 1)gk xk (ξ ), ϕk xk (ξ ) ,

(45)

k=n m−1 



ϕm xm (ξ ) = A(m, n)ϕn (ξ ) + A(m, k + 1)hk xk (ξ ), ϕk xk (ξ ) ,

(46)

k=n

for every ξ ∈ En and every m > n, where



F(m, n) ξ, ϕn (ξ ) = xm (ξ ), ϕm xm (ξ ) ∈ Em × Fm , and prove that these solutions verify (43) and (44). Given n ∈ N and C > D (note that C > 1), let B = Bn be the space of sequences of C 1,1 functions xm : En → Em that, for every m  n and ξ, ξ¯ ∈ En , satisfy the following conditions xn (ξ ) = ξ,

xm (0) = 0,

(47)

dξ xm   C(m − n + 1)a nε ,

(48)

dξ xm − dξ¯ xm   C(m − n + 1)a n2ε ξ − ξ¯ .

(49)

From the mean value theorem, (47) and (48) it follows that   xm (ξ )  C(m − n + 1)a nε ξ 

(50)

for every ξ ∈ En . This allows us to equip B with the metric induced by 

xm (ξ )(m − n + 1)−a n−ε : m  n, ξ ∈ En \ {0} x = sup ξ  

 (51)

for x = (xm )mn ∈ B. Note that x  C for each x ∈ B. Proposition 1. The space B is a complete metric space with the metric induced by (51). Proof. Let (xk )k = ((xm,k )mn )k be a Cauchy sequence in B with respect to the metric induced by (51). Then for each m  n the sequence (xm,k |Bn (r) )k∈N is a Cauchy sequence with respect to the supremum norm in the space of bounded functions from Bn (r) into Fm . Here Bn (r) is the open ball of En centered at 0 and with radius r. Therefore, for each m  n, there exists a function ym : Bn (r) → Fm such that (xm,k |Bn (r) )k∈N converges to ym in the space of bounded functions from Bn (r) into Fm equipped with the supremum norm. For every ξ ∈ Bn (r), from (50), (48) and (49) we get

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137

  xm,k (ξ )  C(m − n + 1)a nε r dξ xm,k   C(m − n + 1)a nε , dξ xm,k − dξ¯ xm,k   C(m − n + 1)a n2ε ξ − ξ¯ . Putting b = C(m − n + 1)a nε max{nε , r}, it follows that, for each m  n and k ∈ N, xm,k ∈ Cb1,1 (Bn (r), Fm ). Here Cb1,1 (Bn (r), Fm ) is the space of C 1,1 functions from Bn (r) into Fm such that the norm defined by

u1,1 = max u∞ , du∞ , L(du) , is less or equal than b,  · ∞ is the supremum norm and   u(x) − u(y) L(u) = sup : x, y ∈ X with x = y . x − y It follows from the generalization of Henry’s lemma (see [15, p. 151]) given by Elbialy [14] (for related results see also [16] and [13]) that ym ∈ Cb1,1 (Bn (r), Fm ) and (dξ xm,k )k∈N converges pointwise to dξ y¯m when k → ∞

(52)

for every ξ ∈ Bn (r). By the uniqueness of each function ym in the ball Bn (r), we can obtain a function y¯m ∈ C 1,1 (Em , Fm ) such that y¯m |Bn (r) = ym for each r > 0. Using (52) we can easily verify that (y¯m )m ∈ B. Moreover, since (xk )k is a Cauchy sequence, for each κ > 0 there exists p ∈ N such that if k, m > p then   xq,k (ξ ) − xq,m (ξ )  κ(q − n + 1)a nε ξ  (53) for every q  n and ξ ∈ Em . Letting m → ∞ in (53) we obtain   xq,k (ξ ) − y¯q (ξ )  κ(q − n + 1)a nε ξ , and thus (xk )k converges to y¯ = (y¯q )qn in the space B.

2

Given ϕ = (ϕn )n∈N ∈ X and x = (xm )mn ∈ B we write

∗ ∗ = ϕm ◦ xm and fm∗ (ξ ) = fm xm (ξ ), ϕm (ξ ) . ϕm Lemma 6. For each ϕ ∈ X, n ∈ N, x ∈ B, ξ, ξ¯ ∈ En , and m  n we have   xm (ξ ) − xm (ξ¯ )  C(m − n + 1)a nε ξ − ξ¯ ,   dξ ϕ ∗   C(m − n + 1)a nε , m   dξ f ∗   2Cδm−3ε−1 (m − n + 1)a nε , m   ∗ ϕ (ξ ) − ϕ ∗ (ξ¯ )  C(m − n + 1)a nε ξ − ξ¯ , m m   dξ ϕ ∗ − d ¯ ϕ ∗   2C 2 (m − n + 1)a n2ε ξ − ξ¯ , m ξ m   dξ f ∗ − d ¯ f ∗   7C 2 δm−3ε−1 (m − n + 1)a n2ε ξ − ξ¯ . m ξ m

(54) (55) (56) (57) (58) (59)

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Proof. By the mean value theorem and (48) we have   a ε xm (ξ ) − xm (ξ¯ )  sup d ¯ ¯ ξ +r(ξ¯ −ξ ) xm  · ξ − ξ   C(m − n + 1) n ξ − ξ  r∈[0,1]

and this establishes (54). Eq. (55) follows immediately from (39) and (48). In fact   dξ ϕ ∗   dx

m (ξ )

m

ϕm  · dξ xm   C(m − n + 1)a nε .

By (36), (48) and (55) we have    dξ f ∗   ∂ 1,0 m

∗ (ξ ) fm xm (ξ ),ϕm

−3ε−1

 2Cδm

   · dξ xm  + ∂ 0,1

∗ (ξ ) fm xm (ξ ),ϕm

    · dξ ϕ ∗  m

(m − n + 1) n

a ε

and this proves (56). Using the mean value theorem and (55) we obtain (57). To prove (58) we note that by (40) and (54) we get   dxm (ξ ) ϕm − dxm (ξ¯ ) ϕm   xm (ξ ) − xm (ξ¯ )  C(m − n + 1)a nε ξ − ξ¯ , and, since C > 1, it follows from (48), (39) and (49) that   dξ ϕ ∗ − d ¯ ϕ ∗   dx (ξ ) ϕm − d ¯ ϕm  · dξ xm  + d ¯ ϕm  · dξ xm − d ¯ xm  m m xm (ξ ) xm (ξ ) ξ m ξ  C 2 (m − n + 1)2a n2ε ξ − ξ¯  + C(m − n + 1)a n2ε ξ − ξ¯   2C 2 (m − n + 1)a n2ε ξ − ξ¯ . Finally we are going to prove (59). Writing  λ,μ  λ,μ Fλ,μ = ∂xm (ξ ),ϕ ∗ (ξ ) fm − ∂x (ξ¯ ),ϕ ∗ (ξ¯ ) fm  m

m

m

where λ, μ ∈ {0, 1} and λ + μ = 1, we have      dξ f ∗ − d ¯ f ∗   F1,0 dξ xm  + F0,1 dξ ϕ ∗  + ∂ 1,0 m

ξ m

m

  · dξ xm − d ¯ xm 

   ∗ ∗ . f  · dξ ϕm − dξ¯ ϕm (ξ¯ ),ϕ ∗ (ξ¯ ) m

 + ∂ 0,1 xm

∗ (ξ¯ ) fm xm (ξ¯ ),ϕm

m

Using (37), (54) and (57) we obtain     ∗ ∗ ¯  Fλ,μ  δm−3ε−1 xm (ξ ) − xm (ξ¯ ) + δm−3ε−1 ϕm (ξ ) − ϕm (ξ )  2Cδm−3ε−1 (m − n + 1)a nε ξ − ξ¯ . By the former inequalities, (48), (55), (36), (49) and (58) it follows from (60) that

ξ

(60)

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139

  dξ f ∗ − d ¯ f ∗   4C 2 δm−3ε−1 (m − n + 1)2a n2ε ξ − ξ¯  m ξ m

+ C + 2C 2 δm−3ε−1 (m − n + 1)a n2ε ξ − ξ¯ . This yields (59) and the lemma is proved.

2

Lemma 7. Given δ > 0 sufficiently small, for each ϕ ∈ X and n ∈ N there exists a unique sequence x = x ϕ ∈ B satisfying (45) for every m > n and ξ ∈ En . Proof. We consider an operator J in B defined by (J x)n (ξ ) = ξ and by (J x)m (ξ ) = A(m, n)ξ +

m−1 



A(m, k + 1)gk xk (ξ ), ϕk xk (ξ )

k=n

for m > n. For every m  n, one can easily verify that (J x)m is a function of class C 1 and, from (47), (38) and (35), that (J x)m (0) = 0. From dξ (J x)m = A(m, n)Pn +

m−1 

A(m, k + 1)Pk+1 dξ fk∗

k=n

it follows from (2), (56) and (8) that      m−1   dξ (J x)m   A(m, n)Pn  + A(m, k + 1)Pk+1 dξ f ∗  k

k=n

 D(m − n + 1)a nε +

m−1 

D(m − k)a (k + 1)ε 2Cδ(k − n + 1)a nε k −3ε−1

k=n

 D(m − n + 1)a nε + 2CDδnε

m−1 

(m − k)a (k − n + 1)a (k + 1)ε k −3ε−1

k=n

 D(m − n + 1) n + 2 a ε

1+ε−a

CDδλ−2ε−1 (m − n + 1)a nε .

Choosing δ sufficiently small (independently of ϕ, x, n, m and ξ ) and since C > D we have   dξ (J x)m   C(m − n + 1)a nε . (61) Proceeding in a similar manner, by (2), (59) and (8) it follows that   m−1     A(m, k + 1)Pk+1  · dξ f ∗ − d ¯ f ∗  dξ (J x)m − d ¯ (J x)m   k

ξ

ξ k

k=n



m−1  k=n

D(m − k)a (k + 1)ε 7C 2 δk −3ε−1 (k − n + 1)a n2ε ξ − ξ¯ 

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 7 2ε−a C 2 Dδλ−2ε−1 (m − n + 1)a n2ε ξ − ξ¯  and choosing δ sufficiently small (independently of ϕ, x, n, m, ξ and ξ¯ ) we have   dξ (J x)m − d ¯ (J x)m   C(m − n + 1)a n2ε ξ − ξ¯ . ξ

(62)

Therefore from (61) and (62) we conclude that J (B) ⊂ B. We now show that J is a contraction in B with the metric induced by (51). Let x, y ∈ B. By (36), the mean value theorem and (39), for k  n we have 



 αk := fk xk (ξ ), ϕk xk (ξ ) − fk yk (ξ ), ϕk yk (ξ )    



 δk −3ε−1 xk (ξ ) − yk (ξ ) + ϕk xk (ξ ) − ϕk yk (ξ )     2δk −3ε−1 xk (ξ ) − yk (ξ )  2δk −3ε−1 (k − n + 1)a nε ξ x − y . Using (2), the former inequality and (8) we obtain    m−1  (J x)m (ξ ) − (Jy)m (ξ )  A(m, k + 1)Pk+1 αk k=n



m−1 

D(m − k)a (k + 1)ε 2δk −3ε−1 (k − n + 1)a nε ξ x − y

k=n

 δθ  (m − n + 1)a nε ξ x − y with θ  = 21+ε−a Dλ−2ε−1 . Therefore, J x − Jy  δθ  x − y , and for δ > 0 sufficiently small J is a contraction in B. By Proposition 1, the map J has a unique fixed point x ϕ in B, which is thus the desired sequence. 2 ϕ

For the next lemma we need to represent by (xn,k )kn ∈ Bn the unique sequence given by Lemma 7. Lemma 8. Given δ > 0 sufficiently small, for each ϕ ∈ X the following properties are equivalent: ϕ

(1) for every n ∈ N, m > n, and ξ ∈ En , the identity (46) holds with x = (xn,k )kn ; (2) for every n ∈ N, and ξ ∈ En we have ϕn (ξ ) = −

∞  k=n

ϕ ϕ

A(k + 1, n)−1 hk xn,k (ξ ), ϕk xn,k (ξ ) .

(63)

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141

Proof. We first show that the series in (63) converges. By the mean value theorem, using (36), (39) and (50) we obtain    ϕ 





fk x (ξ ), ϕk x ϕ (ξ )   δk −3ε−1 x ϕ (ξ ) + ϕk x ϕ (ξ )  n,k n,k n,k n,k   ϕ  2δk −3ε−1 x (ξ ) n,k

 2δCk

−3ε−1

(k − n + 1)a nε ξ .

From (3), we conclude that ∞  

 A(k + 1, n)−1 hk x ϕ (ξ ), ϕk x ϕ (ξ )  n,k n,k k=n

=

∞  

 A(k + 1, n)−1 Qk+1 fk x ϕ (ξ ), ϕk x ϕ (ξ )  n,k n,k k=n



∞ 

D(k − n + 2)−b (k + 1)ε 2δCk −3ε−1 (k − n + 1)a nε ξ 

k=n

 21+ε δCDnε ξ 

∞ 

(k − n + 1)a−b k −2ε−1

k=n

 21+ε δCDnε ξ 

∞ 

(k − n + 1)a−b−2ε−1 < ∞.

k=n

If the first property holds, the identity A(m, n)−1 A(m, k + 1) = A(k + 1, n)−1 allows to write (46) in the following equivalent form  ϕ ϕ ϕ

m−1

(ξ ) − A(k + 1, n)−1 hk xn,k (ξ ), ϕk xn,k (ξ ) . ϕn (ξ ) = A(m, n)−1 ϕm xn,m

(64)

k=n

By (3) and (39), it follows that 

 

 A(m, n)−1 ϕm x ϕ (ξ )  = A(m, n)−1 Qm ϕm x ϕ (ξ )  n,m n,m   ϕ  D(m − n + 1)−b mε xn,m (ξ )  D(m − n − 1)−b mε C(m − n + 1)a nε ξ   CD(m − n + 1)a−b+ε n2ε ξ . Therefore by (42) we have A(m, n)−1 ϕm (xn,m (ξ )) → 0 when m → ∞ and letting m → ∞ in (64) we obtain (63). We now assume that (63) holds. Then ϕ

A(m, n)ϕn (ξ ) = −

∞  k=n

ϕ ϕ

A(m, k + 1)hk xn,k (ξ ), ϕk xn,k (ξ ) ,

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and thus A(m, n)ϕn (ξ ) +

m−1 

ϕ ϕ

A(m, k + 1)hk xn,k (ξ ), ϕk xn,k (ξ )

k=n

=−

∞ 

ϕ ϕ

A(m, k + 1)hk xn,k (ξ ), ϕk xn,k (ξ ) .

(65)

k=m ϕ

It follows from (63) that the right-hand side of (65) is ϕm (xn,m (ξ )), and Eq. (46) is satisfied with x = xϕ . 2 We now equip the space X with metric induced by   ϕn (x) : n ∈ N and x ∈ En \ {0} . ϕ = sup x

(66)

Proposition 2. The space X is a complete metric space with the metric induced by (66). The proof of Proposition 2 is completely analogous to the proof of Proposition 1 and thus it is omitted. Lemma 9. Given δ > 0 sufficiently small, for each ϕ, ψ ∈ X, n ∈ N, m  n, and ξ ∈ En we have  ϕ  x (ξ ) − x ψ (ξ )  C (m − n + 1)a nε ξ  · ϕ − ψ . m m 2 Proof. Putting  ϕ ϕ

ψ ψ

 γk = fk xk (ξ ), ϕk xk (ξ ) − fk xk (ξ ), ψk xk (ξ ) ,

(67)

by the mean value theorem and (36) we obtain   ϕ

 ϕ ψ 

ψ γk  δk −3ε−1 xk (ξ ) − xk (ξ ) + ϕk xk (ξ ) − ψk xk (ξ )  . Using again the mean value theorem, (39) and (50) we have  ϕ



 

 

 ϕk x (ξ ) − ψk x ψ (ξ )   ϕk x ϕ (ξ ) − ψk x ϕ (ξ )  + ψk x ϕ (ξ ) − ψk x ψ (ξ )  k k k k k k  ϕ   ϕ  ψ  ϕ − ψ · xk (ξ ) + xk (ξ ) − xk (ξ )  ϕ  ψ  C(k − n + 1)a nε ϕ − ψ ξ  + xk (ξ ) − xk (ξ ) and this implies that   ϕ

ψ γk  δk −3ε−1 2xk (ξ ) − xk (ξ ) + C(k − n + 1)a nε ϕ − ψ ξ  .

(68)

We are now going to prove the lemma by induction. For k = n the result follows immediately. Suppose that the result is true for k = n, . . . , m − 1. Then for k = n, . . . , m − 1 we get from (68)

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143

and the induction hypothesis that

γk  δk −3ε−1 C(k − n + 1)a nε ϕ − ψ ξ  + C(k − n + 1)a nε ϕ − ψ ξ   2δC(k − n + 1)a nε k −3ε−1 ϕ − ψ ξ . Hence by (2), the last inequality and (8) it follows that   m−1   ϕ A(m, k + 1)Pk+1 γk x (ξ ) − x ψ (ξ )  m m k=n



m−1 

D(m − k)a (k + 1)ε 2δC(k − n + 1)a nε k −3ε−1 ϕ − ψ ξ 

k=n

 21+ε−a CDδλ−2ε−1 (m − n + 1)a nε ϕ − ψ ξ . Choosing δ such that 21+ε−a Dδλ−2ε−1 < 1/2 we have   ϕ x (ξ ) − x ψ (ξ )  C (m − n + 1)a nε ϕ − ψ ξ  m m 2 and this concludes the proof of the lemma.

2

Lemma 10. Given δ > 0 sufficiently small, there is a unique ϕ ∈ X such that (63) holds for every n ∈ N and ξ ∈ En . Proof. We consider the operator Φ defined for each ϕ ∈ X by (Φϕ)n (ξ ) = −

∞ 

ϕ ϕ

A(k + 1, n)−1 hk xk (ξ ), ϕk xk (ξ ) ,

k=n ϕ

where x ϕ = (xk )kn is the unique sequence given by Lemma 7. One can easily verify that each ϕ function (Φϕ)n is of class C 1 . Since x ϕ ∈ B we have xm (0) = 0, m  n. It follows from (38) and (35) that (Φϕ)n (0) = 0 for each n ∈ N. We also have d0 (Φϕ)n = 0 for each n ∈ N because ϕ d0 fk = 0, k ∈ N, ϕm (0) = 0 and xm (0) = 0, m  n. Furthermore, by (3) and (56) we have ∞        A(k + 1, n)−1 Qk+1  · dξ f ∗  dξ (Φϕ)n   k

k=n

 2CDδ

∞  (k − n + 2)−b (k + 1)ε k −3ε−1 (k − n + 1)a nε k=n

 21+ε CDδ

∞  (k − n + 2)−b (k − n + 1)a k −ε−1 k=n

 21+ε CDδ

∞  (k − n + 1)a−b−ε−1 < ∞. k=n

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Hence, for δ > 0 sufficiently small (independently of ϕ, n and ξ ) we have dξ (Φϕ)n   1 for every n ∈ N. On the other hand, given ξ, ξ¯ ∈ En it follows from (59) that ∞        dξ (Φϕ)n − d ¯ (Φϕ)n   A(k + 1, n)−1 Qk+1  · dξ f ∗ − d ¯ f ∗  k ξ ξ k k=n

 7C 2 Dδn2ε ξ − ξ¯ 

∞  (k − n + 2)−b (k + 1)ε k −3ε−1 (k − n + 1)a k=n

 7 2ε C 2 Dδξ − ξ¯ 

∞  (k − n + 1)a−b−1 k=n

 ξ − ξ¯  provided that δ > 0 is sufficiently small (independently of ϕ, n, ξ and ξ¯ ). Hence, Φ(X) ⊂ X. We now show that Φ is a contraction. Given ϕ, ψ ∈ X and n ∈ N, let x ϕ and x ψ be the unique sequences given by Lemma 7 respectively for ϕ and ψ . By (68) (see (67) for the definition of γk ) and Lemma 9 we have   (Φϕ)n (ξ ) − (Φψ)n (ξ ) 

∞    A(k + 1, n)−1 Qk+1 γk k=n



∞    A(k + 1, n)−1 Qk+1 δk −3ε−1 k=n

× C(k − n + 1)a ξ  · ϕ − ψ + C(k − n + 1)a nε ξ  · ϕ − ψ

∞   2CDδξ  · ϕ − ψ (k − n + 2)−b (k + 1)ε k −3ε−1 (k − n + 1)a nε 

k=n

 21+ε CDδξ  · ϕ − ψ

∞  (k − n + 1)a−b−ε−1 . k=n

Therefore, choosing δ > 0 such that λ := 21+ε CDδλa−b−ε−1 < 1, it follows that Φϕ − Φψ  λϕ − ψ , and with this λ we obtain a contraction in X. By Proposition 2, the map Φ has a unique fixed point ϕ in X that is the desired sequence. 2 Proof of Theorem 2. By Lemma 7, for each ϕ ∈ X there is a unique sequence x ϕ ∈ B satisfying identity (45). It remains to solve the identity (46) with x = x ϕ . By Lemma 8, this is the same as solving (63). Finally, by Lemma 10, there is a unique solution of (63). This establishes the existence of the stable manifolds for δ > 0 sufficiently small. Since the functions ϕn are of class

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C 1,1 the graphs Vn are C 1,1 manifolds. Finally, for each n ∈ N, m  n, ξ, ξ¯ ∈ En , it follows from (54) and (57) that       F(m, n)(v) − F(m, n)(v) ¯   xm (ξ ) − xm (ξ¯ ) + ϕ ∗ (ξ ) − ϕ ∗ (ξ¯ ) m

m

 2C(m − n + 1) n ξ − ξ¯  a ε

and from (49) and (58) that     dv F(m, n) − dv¯ F(m, n)  dξ xm − d ¯ xm  + dξ ϕ ∗ − d ¯ ϕ ∗  m ξ ξ m

 C + 2C 2 (m − n + 1)a n2ε ξ − ξ¯ , where v = (ξ, ϕn (ξ )) and v¯ = (ξ¯ , ϕn (ξ¯ )). This completes the proof of the theorem.

2

5. Examples In this section we are going to give examples of nonuniform polynomial dichotomies and two families of perturbations, the first one verifying our assumptions in Section 3 and the second one verifying the assumptions of Section 4. Let a < 0  b and ε  0. We are going to construct a sequence of linear operators An : R2 → R2 given by diagonal matrices 

a An = n 0

0 bn



with positive entries in the diagonal and that verify (2) and (3) with D = 1 and projections given by Pn (x, y) = (x, 0) and Qn (x, y) = (0, y). Because A(2, 1)P1  = a1 , from (2) we must have a1  2a . Therefore we put a1 = 2a . Using again (2) we must have   A(3, 2)P2   2a 2ε

  and A(3, 1)P1   3a .

Since A(3, 2)P2  = a2 and A(3, 1)P1  = a2 a1 , we put   3a . a2 = min 2a 2ε , a1 Using the same arguments we put   3a 2ε 4a a3 = min 2a 3ε , . , a2 a2 a1 Hence the values of an are given recursively by a1 = 2a and   3a n ε na 2ε (n + 1)a ,..., , an+1 = min 2a (n + 1)ε , . an a n . . . a2 a n . . . a 1

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Using similar arguments, from (3) we set b1 = 2b 2−ε and  bn+1 = max

 2b 3b nb (n + 1)b . , , . . . , , (n + 1)ε (n + 1)ε bn (n + 1)ε bn . . . b2 (n + 1)ε bn . . . b1

It follows immediately from the construction that (An ) admits a nonuniform polynomial dichotomy. For instance, if ε = −a = b we can easily verify that 

a ( n+1 n ) An = 0

 0 . 1

For the local case considered in Section 3 we have the perturbations fn : R2 → R2 given by

fn (x, y) = x q+1 , y q+1

or fn (x, y) = y q+1 , x q+1

with q ∈ N. Obviously we have fn (0, 0) = (0, 0) and from inequalities  √  a 2 + b2  |a| + |b|  2 a 2 + b2 and   q q       q+1

q  q+1 q−k k a − b  = (a − b) a b   |a − b| |a|q−k |b|k  |a − b| |a| + |b|   k=1

k=1

it follows that   



fn (x, y) − fn (u, v) = x q+1 − uq+1 2 + y q+1 − v q+1 2      x q+1 − uq+1  + y q+1 − v q+1 

q

q  |x − u| |x| + |u| + |y − v| |y| + |v|



q  |x − u| + |y − v| |x| + |u| + |y| + |v| √  

q  2(x, y) − (u, v) |x| + |u| + |y| + |v|  q       2(q+1)/2 (x, y) − (u, v) (x, y) + (u, v) . This proves that the family of perturbations fn satisfy (10) and (11). Therefore if we have aq + ε + 1 < 0 and a + β < 0, using Theorem 1 with (An ) and (fn ) as above, we get local Lipschitz stable manifolds for the dynamics defined by (4). 2 Let g : R → R be the function defined by g(t) = t 2 e−t . It is easy to see that     

g (t) = 2t 1 − t 2 e−t 2   1

(69)

   g (t) − g  (s)  2|t − s|

(70)

and

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for every t ∈ R. Then if fn : R2 → R2 is the function defined by

δ fn (x, y) = n−3ε−1 g(x), g(y) 2



δ or fn (x, y) = n−3ε−1 g(y), g(x) , 2

from (69) and (70) it follows immediately that the functions fn satisfy (35), (36) and (37). A closer look in the proofs of Lemmas 7, 9 and 10, allows us to conclude, taking C = 2D, that it is sufficient to have δ<

1 , 14 2ε−a D 3 λ−ε−1

and in our example, since D = 1 and λ−ε−1 < 1 + 1/ε, it is enough to have δ<

1 14 2ε−a (1 + 1/ε)

.

With these (fn ) and with (An ) as above, using Theorem 2 we conclude that the dynamics given by (4) has global stable C 1 manifolds when a + ε < b and ε > 0. References [1] L. Barreira, Y. Pesin, Lyapunov Exponents and Smooth Ergodic Theory, Univ. Lecture Ser., vol. 23, American Mathematical Society, Providence, RI, 2002. [2] L. Barreira, Y. Pesin, Nonuniform Hyperbolicity: Dynamics of Systems with Nonzero Lyapunov Exponents, Encyclopedia Math. Appl., vol. 115, Cambridge University Press, Cambridge, 2007. [3] L. Barreira, C. Silva, C. Valls, Integral stable manifolds in Banach spaces, J. Lond. Math. Soc. (2) 77 (2) (2008) 443–464. [4] L. Barreira, C. Silva, C. Valls, Regularity of invariant manifolds for nonuniformly hyperbolic dynamics, J. Dynam. Differential Equations 20 (2) (2008) 281–299. [5] L. Barreira, C. Valls, Higher regularity of invariant manifolds for nonautonomous equations, Nonlinearity 18 (5) (2005) 2373–2390. [6] L. Barreira, C. Valls, Smoothness of invariant manifolds for nonautonomous equations, Comm. Math. Phys. 259 (3) (2005) 639–677. [7] L. Barreira, C. Valls, Existence of stable manifolds for nonuniformly hyperbolic C 1 dynamics, Discrete Contin. Dyn. Syst. 16 (2) (2006) 307–327. [8] L. Barreira, C. Valls, Smooth invariant manifolds in Banach spaces with nonuniform exponential dichotomy, J. Funct. Anal. 238 (1) (2006) 118–148. [9] L. Barreira, C. Valls, Stable manifolds for nonautonomous equations without exponential dichotomy, J. Differential Equations 221 (1) (2006) 58–90. [10] L. Barreira, C. Valls, Stability theory and Lyapunov regularity, J. Differential Equations 232 (2) (2007) 675–701. [11] L. Barreira, C. Valls, Characterization of stable manifolds for nonuniform exponential dichotomies, Discrete Contin. Dyn. Syst. 21 (4) (2008) 1025–1046. [12] L. Barreira, C. Valls, Growth rates and nonuniform hyperbolicity, Discrete Contin. Dyn. Syst. 22 (3) (2008) 509– 528. [13] S.-N. Chow, K. Lu, C k centre unstable manifolds, Proc. Roy. Soc. Edinburgh Sect. A 108 (3–4) (1988) 303–320. [14] M.S. Elbialy, On sequences of Cbk,δ maps which converges in the uniform C 0 -norm, Proc. Amer. Math. Soc. 128 (11) (2000) 3285–3290. [15] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, Springer-Verlag, Berlin, 1981. [16] O.E. Lanford III, Bifurcation of periodic solutions into invariant tori: The work of Ruelle and Takens, in: I. Stakgold, D.D. Joseph, D.H. Sattinger (Eds.), Nonlin. Probl. Phys. Sci. Biology, Proc. Battelle Summer Inst., Seattle, 1972, in: Lecture Notes in Math., vol. 322, Springer, Berlin, 1973, pp. 159–192.

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[17] R. Mañé, Lyapunov exponents and stable manifolds for compact transformations, in: J. Palis Jr. (Ed.), Geometric Dynamics, Rio de Janeiro, 1981, in: Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 522–577. [18] V.I. Oseledets, A multiplicative ergodic theorem. Ljapunov characteristic numbers for dynamical systems, Tr. Mosk. Mat. Obs. 19 (1968) 179–210 (in Russian); English transl.: Trans. Moscow Math. Soc. 19 (1968) 197–231. [19] O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z. 32 (1) (1930) 703–728. [20] Y. Pesin, Families of invariant manifolds that corresponding to nonzero characteristic exponents, Izv. Akad. Nauk SSSR Ser. Mat. 40 (6) (1976) 1332–1379 (in Russian); English transl.: Math. USSR-Izv. 10 (1976) 1261–1305. [21] Y. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspekhi Mat. Nauk 32 (4) (1977) 55–112 (in Russian); English transl.: Russian Math. Surveys 32 (1977) 55–114. [22] Y. Pesin, Geodesic flows in closed Riemannian manifolds without focal points, Izv. Akad. Nauk SSSR Ser. Mat. 41 (6) (1977) 1252–1288 (in Russian); English transl.: Math. USSR-Izv. 11 (1977) 1195–1228. [23] C. Pugh, M. Shub, Ergodic attractors, Trans. Amer. Math. Soc. 312 (1) (1989) 1–54. [24] D. Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math. 50 (1) (1979) 27–58. [25] D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2) 115 (2) (1982) 243–290. [26] G.R. Sell, Y. You, Dynamics of Evolutionary Equations, Appl. Math. Sci., vol. 143, Springer-Verlag, New York, 2002. [27] P. Thieullen, Fibrés dynamiques asymptotiquement compacts. Exposants de Lyapounov. Entropie. Dimension, Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1) (1987) 49–97.

Journal of Functional Analysis 257 (2009) 149–193 www.elsevier.com/locate/jfa

Banach spaces without minimal subspaces Valentin Ferenczi a , Christian Rosendal b,∗,1 a Departamento de Matemática, Instituto de Matemática e Estatística, Universidade de São Paulo,

05311-970 São Paulo, SP, Brazil b Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago,

322 Science and Engineering Offices (M/C 249), 851 S. Morgan Street, Chicago, IL 60607-7045, USA Received 21 October 2008; accepted 23 January 2009 Available online 20 February 2009 Communicated by K. Ball

Abstract We prove three new dichotomies for Banach spaces à la W.T. Gowers’ dichotomies. The three dichotomies characterise respectively the spaces having no minimal subspaces, having no subsequentially minimal basic sequences, and having no subspaces crudely finitely representable in all of their subspaces. We subsequently use these results to make progress on Gowers’ program of classifying Banach spaces by finding characteristic spaces present in every space. Also, the results are used to embed any partial order of size ℵ1 into the subspaces of any space without a minimal subspace ordered by isomorphic embeddability. © 2009 Elsevier Inc. All rights reserved. Keywords: Minimal Banach spaces; Dichotomies; Classification of Banach spaces

Contents 1. 2.

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Introduction . . . . . . . . . . . . . . . . . . . . . . . Preliminaries . . . . . . . . . . . . . . . . . . . . . . . 2.1. Notation, terminology, and conventions 2.2. Gowers’ block sequence game . . . . . . 2.3. A trick and a lemma . . . . . . . . . . . . . Tightness . . . . . . . . . . . . . . . . . . . . . . . . .

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* Corresponding author.

E-mail addresses: [email protected] (V. Ferenczi), [email protected] (C. Rosendal). 1 Partially supported by NSF grant DMS 0556368 and by FAPESP.

0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.01.028

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3.1. Tight bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. A generalised asymptotic game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. A game for minimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. A dichotomy for minimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Tightness with constants and crude stabilisation of local structure . . . . . . . . . . . . . . 5. Local block minimality, asymptotic structure and a dichotomy for containing c0 or p 6. Tightness by range and subsequential minimality . . . . . . . . . . . . . . . . . . . . . . . . . 7. Chains and strong antichains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Refining Gowers’ dichotomies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Open problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1. Introduction In the paper [20], W.T. Gowers initiated a celebrated classification theory for Banach spaces. Since the task of classifying all (even separable) Banach spaces up to isomorphism is extremely complicated (just how complicated is made precise in [11]), one may settle for a loose classification of Banach spaces up to subspaces, that is, look for a list of classes of Banach spaces such that: (a) each class is pure, in the sense that if a space belongs to a class, then every subspace belongs to the same class, or maybe, in the case when the properties defining the class depend on a basis of the space, every block subspace belongs to the same class, (b) the classes are inevitable, i.e., every Banach space contains a subspace in one of the classes, (c) any two classes in the list are disjoint, (d) belonging to one class gives a lot of information about operators that may be defined on the space or on its subspaces. We shall refer to this list as the list of inevitable classes of Gowers. Many classical problems are related to this classification program, as for example the question whether every Banach space contains a copy of c0 or p , solved in the negative by B.S. Tsirelson in 1974 [42], or the unconditional basic sequence problem, also solved negatively by Gowers and B. Maurey in 1993 [21]. Ultimately one would hope to establish such a list so that any classical space appears in one of the classes, and so that belonging to that class would yield most of the properties which are known for that space. For example, any property, which is known for Tsirelson’s space, is also true for any of its block subspaces. So Tsirelson’s space is a pure space, and, as such, should appear in one of the classes with a reasonable amount of its properties. Also, presumably the nicest among the classes would consist of the spaces isomorphic to c0 or p , 1  p < ∞. After the discovery by Gowers and Maurey of the existence of hereditarily indecomposable (or HI) spaces, i.e., spaces such that no subspace may be written as the direct sum of infinite-dimensional subspaces [21], Gowers proved that every Banach space contains either an HI subspace or a subspace with an unconditional basis [19]. These were the first two examples of inevitable classes. We shall call this dichotomy the first dichotomy of Gowers. He then used his famous Ramsey or determinacy theorem [20] to refine the list by proving that any Banach space contains a subspace with a basis such that either no two disjointly supported block subspaces are isomorphic, or such that any two subspaces have further subspaces which are isomorphic. He

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called the second property quasi minimality. This second dichotomy divides the class of spaces with an unconditional basis into two subclasses (up to passing to a subspace). Finally, recall that a space is minimal if it embeds into any of its subspaces. A quasi minimal space which does not contain a minimal subspace is called strictly quasi minimal, so Gowers again divided the class of quasi minimal spaces into the class of strictly quasi minimal spaces and the class of minimal spaces. Obviously the division between minimal and strictly quasi minimal spaces is not a real dichotomy, since it does not provide any additional information. The main result of this paper is to provide the missing dichotomy for minimality, which we shall call the third dichotomy. A first step in that direction was obtained by A.M. Pełczar, who showed that any strictly quasi minimal space contains a further subspace with the additional property of not containing any subsymmetric sequence [35]. The first author proved that the same holds if one replaces subsymmetric sequences by embedding homogeneous sequences (any subspace spanned by a subsequence contains an isomorphic copy of the whole space) [10]. A crucial step in the proofs of [35] and [10] is the notion of asymptoticity. An asymptotic game of length k in a space E with a basis is a game where I plays integers ni and II plays block vectors xi supported after ni , and where the outcome is the length k sequence (xi ). Asymptotic games have been studied extensively and the gap between finite-dimensional and infinite-dimensional phenomena was usually bridged by fixing a constant and letting the length of the game tend to infinity. For example, a basis is asymptotic p if there exists C such that for any k, I has a winning strategy in the length k asymptotic game so that the outcome is C-equivalent to the unit vector basis of kp . In [35] it is necessary to consider asymptotic games of infinite length, which are defined in an obvious manner. The outcome is then an infinite block sequence. The proof of the theorem in [35] is based on the obvious fact that if a basic sequence (ei ) is subsymmetric and (xi ) is a block sequence of (ei ), then II has a strategy in the infinite asymptotic game in E = [ei ] to ensure that the outcome is equivalent to (xi ). In [10] a similar fact for embedding homogeneous basic sequences is obtained, but the proof is more involved and a more general notion of asymptoticity must be used. Namely, a generalised asymptotic game in a space E with a basis (ei ) is a game where I plays integers ni and II plays integers mi and vectors xi such that supp(xi ) ⊆ [n1 , m1 ] ∪ · · · ∪ [ni , mi ], and the outcome is the sequence (xi ), which may no longer be a block basis. The second author analysed infinite asymptotic games in [38] (a previous study had also been undertaken by E. Odell and T. Schlumprecht in [34]), showing that the most obvious necessary conditions are, in fact, also sufficient for II to have a strategy to play inside a given set. This was done through relating the existence of winning strategies to a property of subspaces spanned by vectors of the basis with indices in some intervals of integers. Now the methods of [38] extend to the setting of generalised asymptotic games and motivate the following definition. A space Y is tight in a basic sequence (ei ) if there is a sequence of successive intervals I0 < I1 < I2 < · · · of N such that for all infinite subsets A ⊆ N, we have      / Y  en  n ∈ Ii , i∈A

where Y  X denotes that Y embeds into X. In other words, any embedding of Y into [ei ] has a “large” image with respect to subsequences of the basis (ei ) and cannot avoid an infinite number of the subspaces [en ]n∈Ii . We then define a tight basis as a basis such that every subspace is tight in it and a tight space as a space with a tight basis.

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As we shall prove in Lemma 3.7, using the techniques of [38], essentially a block subspace Y = [yi ] is not tight in (ei ), when II has a winning strategy in the generalised asymptotic game in [ei ] for producing a sequence equivalent to (yi ). This relates the notion of tight bases to the methods of [10], and by extending these methods we prove the main result of this paper: Theorem 1.1 (3rd dichotomy). Let E be a Banach space without minimal subspaces. Then E has a tight subspace. Theorem 1.1 extends the theorems of [35,10], since it is clear that a tight space cannot contain a subsymmetric or even embedding homogeneous block sequence. This dichotomy also provides an improvement to the list of Gowers: a strictly quasi minimal space must contain a tight quasi minimal subspace. Example 3.6 shows that this is a non-trivial refinement of the unconditional and strictly quasi minimal class, and Corollary 4.3 states that Tsirelson’s space is tight. Theorem 1.1 also refines the class of HI spaces in the list, i.e., every HI space contains a tight subspace, although it is unknown whether the HI property for a space with a basis does not already imply that the basis is tight. Our actual examples of tight spaces turn out to satisfy one of two stronger forms of tightness. The first is called tightness with constants. A basis (en ) is tight with constants when for every infinite-dimensional space Y , the sequence of successive intervals I0 < I1 < · · · of N witnessing / IK ] for each K. This is the case the tightness of Y in (en ) may be chosen so that Y K [en | n ∈ for Tsirelson’s space. The second kind of tightness is called tightness by range. Here the range, range x, of a vector x is thesmallest interval of integers containing its support, and the range of a block subspace [xn ] is n range xn . A basis (en ) is tight by range when for every block subspace Y = [yn ], the sequence of successive intervals I0 < I1 < · · · of N witnessing the tightness of Y in (en ) may be defined by Ik = range yk for each k. This is equivalent to no two block subspaces with disjoint ranges being comparable. In a companion paper [14], we show that tightness by range is satisfied by an HI space and also by a space with unconditional basis both constructed by Gowers. It turns out that there are natural dichotomies between each of these strong forms of tightness and respective weak forms of minimality. For the first notion, we define a space X to be locally minimal if for some constant K, X is K-crudely finitely representable in any of its subspaces. Notice that local minimality is easily incompatible with tightness with constants. Using an equivalent form of Gowers’ game, as defined by J. Bagaria and J. López-Abad [4], we prove: Theorem 1.2 (5th dichotomy). Any Banach space E contains a subspace with a basis that is either tight with constants or is locally minimal. The ideas involved in the notion of local minimality also make sense for block representability, which allows us to connect these notions with asymptoticity of basic sequences. Proving a simple dichotomy for when a space contains an asymptotically p subspace, we are led to the following dichotomy for when a Banach space contains a copy of either c0 or p . Theorem 1.3 (The c0 and p dichotomy). Suppose X is a Banach space not containing a copy of c0 nor of p , 1  p < ∞. Then X has a subspace Y with a basis such that either

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(1) ∀M ∃n ∀U1 , . . . , U2n ⊆ Y ∃ui ∈ SUi 

 u1 < · · · < u2n & (u2i−1 )ni=1 M (u2i )ni=1 .

(2) For all block bases (zn ) of Y = [yn ] there are intervals I1 < I2 < I3 < · · · such that (zn )n∈IK is not K-equivalent to a block sequence of (yn )n∈I / K. Here, of course, the variables range over infinite-dimensional spaces. Property (1) indicates some lack of homogeneity and (2) some lack of minimality. It is interesting to see which conditions the various examples of Banach spaces not containing c0 or p satisfy; obviously, Tsirelson’s space and its dual satisfy (2) and indeed (2) is the only option for spaces being asymptotic p . On the other hand, Schlumprecht’s space S satisfies (1). There is also a dichotomy concerning tightness by range. This direction for refining the list of inevitable classes of spaces was actually suggested by Gowers in [20]. P. Casazza proved that if a space X has a shrinking basis such that no block sequence is even–odd (the odd subsequence is equivalent to the even subsequence), then X is not isomorphic to a proper subspace, see [17]. So any Banach space contains either a subspace, which is not isomorphic to a proper subspace, or is saturated with even–odd block sequences, and, in the second case, we may find a further subspace in which Player II has a winning strategy to produce even–odd sequences in the game of Gowers associated to his Ramsey theorem. This fact was observed by Gowers, but it was unclear to him what to deduce from the property in the second case. We answer this question by using Gowers’ theorem to obtain a dichotomy which on one side contains tightness by range, which is a slightly stronger property than the Casazza property. On the other side, we define a space X with a basis (xn ) to be subsequentially minimal if every subspace of X contains an isomorphic copy of a subsequence of (xn ). This last property is satisfied by Tsirelson’s space and will also be shown to be incompatible with tightness by range. Theorem 1.4 (4th dichotomy). Any Banach space E contains a subspace with a basis that is either tight by range or is subsequentially minimal. It is easy to check that the second case in Theorem 1.4 may be improved to the following hereditary property of a basis (xn ), that we call sequential minimality: every block sequence of [xn ] has a further block sequence (yn ) such that every subspace of [xn ] contains a copy of a subsequence of (yn ). The five dichotomies and the interdependence of the properties involved can be visualised in the following diagram. Unconditional basis ⇑ Tight by support ⇓ Tight by range ⇓ Tight ⇑ Tight with constants

∗ ∗ 1st dichotomy ∗ ∗ ∗ ∗ 2nd dichotomy ∗ ∗ ∗ ∗ 4th dichotomy ∗ ∗ ∗ ∗ 3rd dichotomy ∗ ∗ ∗ ∗ 5th dichotomy ∗ ∗

Hereditarily indecomposable ⇓ Quasi minimal ⇑ Sequentially minimal ⇑ Minimal ⇓ Locally minimal

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From a different point of view, coming from combinatorics and descriptive set theory, Theorem 1.1 also has important consequences for the isomorphic classification of separable Banach spaces. To explain this, suppose that X is a Banach space and SB∞ (X) is the class of all infinitedimensional subspaces of X. Then the relation  of isomorphic embeddability induces a partial order on the set of biembeddability classes of SB∞ (X) and we denote this partial order by P(X). Many questions about the isomorphic structure of X translate directly into questions about the structure of P(X), e.g., X has a minimal subspace if and only if P(X) has a minimal element and X is quasi minimal if and only if P(X) is downwards directed. In some sense, a space can be said to be pure in case the complexity of P(X) does not change by passing to subspaces and Gowers [20, Problem 7.9], motivated by this, asked for a classification of, or at least strong structural information about, the partial orders P for which there is a Banach space X saturated with subspaces Y ⊆ X such that P ∼ = P(Y ). A simple diagonalisation easily shows that such P either consist of a single point (corresponding to a minimal space) or are uncountable, and, using methods of descriptive set theory and metamathematics, this was successively improved in [12] and [37] to either |P | = 1 or P having a continuum size antichain. Using a strengthening of Theorem 1.1, we are now able to show that such P , for which |P | > 1, have an extremely complex structure by embedding any partial order of size at most ℵ1 into them. For A, B ⊆ N, we write A ⊆∗ B to mean that A \ B is finite. Theorem 1.5. Given a Banach space X, let P(X) be the set of all biembeddability classes of infinite-dimensional subspaces of X, partially ordered under isomorphic embeddability. Let P be a poset for which there exists a Banach space X such that X is saturated with subspaces Y such that P(Y ) ∼ = P . Then either |P | = 1, or ⊆∗ embeds into P . In the second case it follows that (a) any partial order of size at most ℵ1 embeds into P , and (b) any closed partial order on a Polish space embeds into P . From the point of view of descriptive set theory, it is more natural to study another problem, part of which was originally suggested to us by G. Godefroy some time ago. Namely, the space SB∞ (X), for X separable, can easily be made into a standard Borel space using the Effros– Borel structure. In this way, the relations of isomorphism, ∼ =, and isomorphic embeddability, , become analytic relations on SB∞ (X) whose complexities can be measured through the notion of Borel reducibility. We obtain Theorem 1.5 as a consequence of some finer results formulated in this language and that are of independent interest. In Section 8, we put all the dichotomies together in order to make progress on the loose classification mentioned above. In connection with this, we shall also rely on work by A. Tcaciuc [41], who proved a dichotomy for containing a strongly asymptotic p basis, i.e., a basis such that finite families of disjointly supported (but not necessarily successive) normalised blocks supported “far enough” are uniformly equivalent to the basis of np . Using just the first four dichotomies, in Theorem 8.3 we find 6 classes of inevitable spaces, 4 of which are known to be non-empty, while if we use all 5 dichotomies plus Tcaciuc’s, we find 19 classes. Out of these, 8 of them are known to be non-empty, though for 4 of the examples, we will need the results of a companion paper [14] where these are constructed and investigated. The resulting classification gives fairly detailed knowledge about the various types of inevitable spaces, though much work remains to be done. In particular, the new dichotomies explains some of the structural differences between the wealth of new exotic spaces constructed

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in the wake of the seminal paper of Gowers and Maurey [21]. It seems an interesting task to determine which of the remaining 11 of the 19 cases are non-empty. 2. Preliminaries 2.1. Notation, terminology, and conventions We shall in the following almost exclusively deal with infinite-dimensional Banach spaces, so to avoid repeating this, we will always assume our spaces to be infinite-dimensional. The spaces can also safely be assumed to be separable, but this will play no role and is not assumed. Moreover, all spaces will be assumed to be over the field of real numbers R, though the results hold without modification for complex spaces too. Suppose E is a Banach space with a normalised Schauder basis (en ). Then, by a standard Skolem hull construction, there is a countable subfield F of R containing the rational numbers Q such that for any finite linear combination λ 0 e0 + λ1 e1 + · · · + λn en with λi ∈ F, we have λ0 e0 + λ1 e1 + · · · + λn en  ∈ F. This means that any F-linear combination of (en ) can be normalised, while remaining a F-linear combination. Thus, as the set of Q and hence also F-linear combinations of (en ) are dense in E, also the set of F-linear normalised combinations of (en ) are dense in the unit sphere SE . A block vector is a normalised finite linear combination x = λ0 e0 + λ1 e1 + · · · + λn en where λi ∈ F. We insist on blocks being normalised and F-linear and will be explicit on the few occasions that we deal with non-normalised blocks. The restriction to F-linear combinations is no real loss of generality, but instead has the effect that there are only countably many blocks. We denote by D the set of blocks. The support, supp x, of a block x = λ0 e0 + λ1 e1 + · · · + λn en is the set of i ∈ N such that λi = 0 and the range, range x, is the smallest interval I ⊆ N containing supp x. A block (sub)sequence, block basis, or blocking of (en ) is an infinite sequence (xn ) of blocks such that supp xn < supp xn+1 for all n and a block subspace is the closed linear span of a block sequence. Notice that if X is a block subspace, then the associated block sequence (xn ) such that X = [xn ] is uniquely defined up to the choice of signs ±xn . So we shall sometimes confuse block sequences and block subspaces. For two block subspaces X = [xn ] and Y = [yn ], write Y  X if Y ⊆ X, or, equivalently, yn ∈ span(xi ) for all n. Also, let Y ∗ X if there is some N such that yn ∈ span(xi ) for all n  N . When we work with block subspaces of some basis (en ), we will assume that we have chosen the same countable subfield F of R for all block sequences (xn ) of (en ), and hence a vector in [xn ] is a block of (xn ) if and only if it is a block of (en ), so no ambiguity occurs. We consider the set bb(en ) of block sequences of (en ) as a closed subset of DN , where D is equipped with the discrete topology. In this way, bb(en ) is a Polish, i.e., separable, completely metrisable space. If  = (δn ) is a sequence of positive real numbers, which we denote by  > 0, and A ⊆ bb(en ), we designate by A the set 

A = (yn ) ∈ bb(en )  ∃(xn ) ∈ bb(en ) ∀n xn − yn  < δn .

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If A is an infinite subset of N, we denote by [A] the space of infinite subsets of A with the topology inherited from 2A . Also, if a ⊆ N is finite,   

[a, A] = B ∈ [N]  a ⊆ B ⊆ a ∪ A ∩ [max a + 1, ∞[ . We shall sometimes confuse infinite subsets of N with their increasing enumeration. So if A ⊆ N is infinite, we denote by An the (n + 1)st element of A in its increasing enumeration (we start counting at 0). A Banach space X embeds into Y if X is isomorphic to a closed subspace of Y . Since we shall work with the embeddability relation as a mathematical object itself, we prefer to use the slightly non-standard notation X  Y to denote that X embeds into Y . Given two Banach spaces X and Y , we say that X is crudely finitely representable in Y if there is a constant K such that for any finite-dimensional subspace F ⊆ X there is an embedding T : F → Y with constant K, i.e., T  · T −1   K. Also, if X = [xn ] and Y = [yn ] are spaces with bases, we say that X is crudely block finitely representable in Y if for some constant K and all k, there are (not necessarily normalised) blocks z0 < · · · < zk of (yn ) such that (x0 , . . . , xk ) ∼K (z0 , . . . , zk ). Two Banach spaces are said to be incomparable if neither one embeds into the other, and totally incomparable if no subspace of one is isomorphic to a subspace of the other. We shall at several occasions use non-trivial facts about the Tsirelson space and its p-convexifications, for which our reference is [8], and also facts from descriptive set theory that can all be found in [24]. For classical facts in Banach space theory we refer to [27]. 2.2. Gowers’ block sequence game A major ingredient in several of our proofs will be the following equivalent version of Gowers’ game due to J. Bagaria and J. López-Abad [4]. Suppose E = [en ] is given. Player I and II alternate in choosing blocks x0 < x1 < x2 < · · · and y0 < y1 < y2 < · · · as follows: Player I plays in the kth round of the game a block xk such that xk−1 < xk . In response to this, II either chooses to pass, and thus play nothing in the kth round, or plays a block yi ∈ [xl+1 , . . . , xk ], where l was the last round in which II played a block. I x0 · · · xk0 xk0 +1 · · · xk1 II y0 ∈ [x0 , . . . , xk0 ] y1 ∈ [xk0 +1 , . . . , xk1 ] We thus see I as constructing a block sequence (xi ), while II chooses a block subsequence (yi ). This block subsequence (yi ) is then called the outcome of the game. (Potentially the blocking could be finite, but the winning condition can be made such that II loses unless it is infinite.) We now have the following fundamental theorem of Gowers (though he only proves it for real scalars, it is clear that his proof is valid for the field F too). Theorem 2.1. (See W.T. Gowers [20].) Suppose (en ) is a Schauder basis and A ⊆ bb(ei ) is an analytic set such that any (xi ) ∈ bb(ei ) has a block subsequence (yi ) belonging to A, then for all  > 0, there is a block subsequence (vi ) ∈ bb(ei ) such that II has a strategy to play in A if I is restricted to play blockings of (vi ).

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2.3. A trick and a lemma We gather here a couple of facts that will be used repeatedly later on. We shall at several occasions use coding with inevitable subsets of the unit sphere of a Banach space, as was first done by López-Abad in [28]. So let us recall here the relevant facts and set up a framework for such codings. Suppose E is an infinite-dimensional Banach space with a basis not containing a copy of c0 . Then by the solution to the distortion problem by Odell and Schlumprecht [31] there is a block subspace [xn ] of E and two closed subsets F0 and F1 of the unit sphere of [xn ] such that dist(F0 , F1 ) = δ > 0 and such that for all block bases (yn ) of (xn ) there are block vectors v and u of (yn ) such that v ∈ F0 and u ∈ F1 . In this case we say that F0 and F1 are positively separated, inevitable, closed subsets of S[xn ] . We can now use the sets F0 and F1 to code infinite binary sequences, i.e., elements of 2N in the following manner. If (zn ) is a block sequence of (xn ) such that for all n, zn ∈ F0 ∪ F1 , we let ϕ((zn )) = α ∈ 2N be defined by  0, if zn ∈ F0 ; αn = 1, if zn ∈ F1 . Since the sets F0 and F1 are positively separated, this coding is fairly rigid and can be extended to block sequences (vn ) such that dist(vn , F0 ∪ F1 ) < 2δ by letting ϕ((vn )) = β ∈ 2N be defined by 0, if dist(vn , F0 ) < 2δ ; βn = 1, if dist(vn , F1 ) < 2δ . In this way we have that if (zn ) and (vn ) are block sequences with zn ∈ F0 ∪F1 and vn −zn  < 2δ for all n, then ϕ((zn )) = ϕ((vn )). One can now use elements of Cantor space 2N to code other objects in various ways. For example, let H denote the set of finite non-empty sequences (q0 , q1 , . . . , qn ) of rationals with qn = 0. Then, as H is countable, we can enumerate it as h0 , h1 , . . . . If now (yn ) and (vn ) are block sequences with ϕ((vn )) = 0n0 10n1 10n2 1 · · · , then (vn ) codes an infinite sequence Ψ ((vn ), (yn )) = (un ) of finite linear combinations of (yn ) by the following rule: uk = q0 y0 + q1 y1 + · · · + qm ym , where hnk = (q0 , . . . , qm ). We should then notice three things about this type of coding: – It is inevitable, i.e., for all block sequences (yn ) of (xn ) and α ∈ 2N , there is a block sequence (vn ) of (yn ) with ϕ((vn )) = α. – It is continuous, i.e., to know an initial segment of (un ) = Ψ ((vn ), (yn )), we only need to know initial segments of (vn ) and of (yn ). – It is stable under small perturbations. I.e., given > 0, we can find some  = (δn ) only depending on and the basis constant of (xn ) with the following property. Assume that (vn ) and (yn ) are block bases of (xn ) with vn ∈ F0 ∪ F1 for all n and such that Ψ ((vn ), (yn )) = (un ) is a block sequence of (yn ) with 12 < un  < 2. Then whenever (vn ) and (yn ) are other

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block sequences of (xn ) with vn − vn  < 2δ and yn − yn  < δn for all n, the sequence Ψ ((vn ), (yn )) = (un ) will be a block sequence of (yn ) that is 1 + -equivalent to (un ). One can of course consider codings of other objects than sequences of vectors and, depending on the coding, obtain similar continuity and stability properties. The inevitability of the coding is often best used in the following form. – Suppose B is a set of pairs ((yn ), α), where (yn ) is a block sequence of (xn ) and α ∈ 2N , such that for all block sequences (zn ) of (xn ) there is a further block sequence (yn ) and an α such that ((yn ), α) ∈ B. Then for all block sequences (zn ) of (xn ) there is a further block sequence (yn ) such that for all n, y2n+1 ∈ F0 ∪ F1 and ((y2n ), ϕ((y2n+1 ))) ∈ B. To see this, let (zn ) be given and notice that by the inevitability of the coding there is a block sequence (wn ) of (zn ) such that w3n+1 ∈ F0 and w3n+2 ∈ F1 . Pick now a block sequence (vn ) of (w3n ) and an α such that ((vn ), α) ∈ B. Notice now that between vn and vn+1 there are block vectors w3in +1 and w3in +2 of (zn ) belonging to F0 , respectively F1 . Thus, if we let y2n = vn and set  w3in +1 , if αn = 0; y2n+1 = w3in +2 , if αn = 1, then ((y2n ), ϕ((y2n+1 ))) ∈ B. Lemma 2.2. Let (xn0 )  (xn1 )  (xn2 )  · · · be a decreasing sequence of block √ bases of a basic 0 0 sequence (xn ). Then there exists a block basis (yn ) of (xn ) such that (yn ) is K-equivalent with a block basis of (xnK ) for every K  1. Proof. Let c(L) be a constant depending on the basis constant of (xn0 ) such that if two block bases differ in at most L terms, then they are c(L)-equivalent. Find √ now a sequence L1  L2  · · · of non-negative integers tending to +∞ such that c(LK )  K. We can now easily construct an infinite block basis (yn ) of (xn0 ) such that for all K  1 at most the first LK terms K of (yn ) are not blocks of (xnK )∞ n=LK +1 . Then (yn ) differs from a block basis of (xn ) in at most √ LK terms and hence is K-equivalent with a block basis of (xnK ). 2 3. Tightness 3.1. Tight bases The following definition is central to the rest of the paper. Definition 3.1. Consider a Banach space E with a basis (en ) and let Y be an arbitrary Banach space. We say that Y is tight in the basis (en ) if there is a sequence of successive non-empty intervals I0 < I1 < I2 < · · · of N such that for all infinite subsets A ⊆ N, we have      / Y  en  n ∈ Ii . i∈A

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In other words, if Y embeds into [en ]n∈B , then B ⊆ N intersects all but finitely many intervals Ii . We say that (en ) is tight if every infinite-dimensional Banach space Y is tight in (en ). Finally, an infinite-dimensional Banach space X is tight if it has a tight basis. Also, the following more analytical criterion will prove to be useful. For simplicity, denote by PI the canonical projection onto [en ]n∈I . Lemma 3.2. Let X be a Banach space, (en ) a basis for a space E, and (In ) finite intervals such that min In − −−− → ∞ and for all infinite A ⊆ N, n→∞ X  [en ]n∈/ k∈A Ik . Then whenever T : X → [en ] is an embedding, we have lim infk PIk T  > 0. Proof. Suppose towards a contradiction that T : X → E is an embedding such that for some infinite A ⊆ N, lim PIk T  = 0.

k→∞ k∈A

Then, by passing to an infinite subset of A, we can suppose that k∈A PIk T  < 12 T −1 −1 and that the intervals (In )n∈A are disjoint. Thus, the sequence of operators (PIk T )k∈A is absolutely

summable and therefore the operator k∈A PIk T : X → E exists and has norm < 12 T −1 −1 . But then for x ∈ X we have          1 1      T −1  · T x = 1 T x,  x  P T x P T Ik Ik  · x     2 2T −1  2T −1  k∈A

k∈A

and hence also             T−    T x − 1 T x = 1 T x.  T x − P T x P T x Ik Ik     2 2 k∈A

k∈A

So T − k∈A PIk T is still an embedding of X into E. But this is impossible as T − k∈A PIk T takes values in [en ]n∈/ k∈A Ik . 2 Proposition 3.3. A tight Banach space contains no minimal subspaces. Proof. Suppose (en ) is a tight basis for a space E and let Y be any subspace of E. Pick a block subspace X = [xn ] of E that embeds into Y . Since Y is tight in (en ), we can find a sequence of intervals (Ii ) such that Y does not embed into [en ]n∈B whenever B ⊆ N is disjoint from an infinite number of intervals Ii . By passing to a subsequence (zn ) of (xn ), we obtain a space Z = [zn ] that is a subspace of some [en ]n∈B where B ⊆ N is disjoint from an infinite number of intervals Ii , and hence Y does not embed into Z. Since Z embeds into Y , this shows that Y is not minimal. 2

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The classical example of space without minimal subspaces is Tsirelson’s space T and it is not too difficult to show that T is tight. This will be proved later on as a consequence of a more general result. Any block sequence of a tight basis is easily seen to be tight. And also: Proposition 3.4. If E is a tight Banach space, then every shrinking basic sequence in E is tight. Proof. Suppose (en ) is a tight basis for E and (fn ) is a shrinking basic sequence in E. Let Y be an arbitrary space and find intervalsI0 < I1 < · · · associated to Y for (en ), i.e., for all infinite subsets A ⊆ N, we have Y  [en | n ∈ / i∈A Ii ]. We notice that, since (en ) is a basis, we have for all m PIk |[fi |im]  − −−→ 0, k→∞

(1)

and, since (fn ) is shrinking and the PIk have finite rank, we have for all k PIk |[fi |i>m]  − −−−→ 0. m→∞

(2)

Using alternately (1) and (2), we can construct integers k0 < k1 < · · · and intervals J0 < J1 < · · · such that PIkn |[fi |i ∈J / n] <

2 . n+1

To see this, suppose kn−1 and Jn−1 have been defined and find some large kn > kn−1 such that PIkn |[fi |imax Jn−1 ]  

1 . n+1

Now, choose m large enough that PIkn |[fi |i>m]  

1 , n+1

2 and set Jn = [max Jn−1 + 1, m]. Then PIkn |[fi |i ∈J / n ]  < n+1 . It follows that if A ⊆ N is infi nite and T : Y → [fi ]i ∈/ n∈A Jn is an embedding, then limn∈A PIkn T  = 0, which contradicts Lemma 3.2. So (Jn ) witnesses that Y is tight in (fn ). 2

Corollary 3.5. If a tight Banach space X is reflexive, then every basic sequence in X is tight. Notice that, since c0 and 1 are minimal, we have by the classical theorem of James, that if X is a tight Banach space with an unconditional basis, then X is reflexive and so every basic sequence in X is tight. Example 3.6. The symmetrisation S(T (p) ) of the p-convexification T (p) of Tsirelson’s space, 1 < p < +∞, does not contain a minimal subspace, yet it is not tight.

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Proof. Since S(T (p) ) is saturated with isomorphic copies of subspaces of T (p) and T (p) does not contain a minimal subspace, it follows that S(T (p) ) does not have a minimal subspace. The canonical basis (en ) of S(T (p) ) is symmetric, therefore S(T (p) ) is not tight in (en ) and so (en ) is not tight. By reflexivity, no basis of S(T (p) ) is tight. 2 3.2. A generalised asymptotic game Suppose X = [xn ] and Y = [yn ] are two Banach spaces with bases. We define the game HY,X with constant C  1 between two players I and II as follows: I will in each turn play a natural number ni , while II will play a not necessarily normalised block vector ui ∈ X and a natural number mi such that ui ∈ X[n0 , m0 ] + · · · + X[ni , mi ], where, for ease of notation, we write X[k, m] to denote [xn ]knm . Diagrammatically, n1 n2 n3 ··· I n0 u1 , m1 u2 , m2 u3 , m3 · · · II u0 , m0 We say that the sequence (ui )i∈N is the outcome of the game and say that II wins the game if (ui ) ∼C (yi ). For simplicity of notation, if X = [xn ] is space with a basis, Y a Banach space, I0 < I1 < I2 < · · · a sequence of non-empty intervals of N and K is a constant, we write Y K (X, Ii ) if there is an infinite set A ⊆ N containing 0 such that      / Ii , Y K xn  n ∈ i∈A

i.e., Y embeds with constant K into the subspace of X spanned by (xn )n∈/ i∈A Ii . Also, write Y  (X, Ii )  if there is an infinite set A ⊆ N such that Y  [xn | n ∈ / i∈A Ii ]. Notice that in the latter case we can always demand that 0 ∈ A by perturbating the embedding with a finite rank operator. It is clear that if Y = [yn ] and II has a winning strategy in the game HY,X with constant K, then for any sequence of intervals (Ii ), Y K (X, Ii ). Modulo the determinacy of open games, the next lemma shows that the converse holds up to a perturbation. Lemma 3.7. Suppose X = [xn ] is space with a basis and K, are positive constants such that for all block bases Y of X there is a winning strategy for I in the game HY,X with constant K + . Then there is a Borel function f : bb(X) → [N] such that for all Y if Ij = [f (Y )2j , f (Y )2j +1 ], then Y K (X, Ij ).

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Proof. Notice that the game HY,X is open for player I and, in fact, if DK+ denotes the set of 1  u  K + , then the set blocks u with K+

 A = (Y, p)  ∈ bb(X) × (N × DK+ × N)N  either p is a legal run of the game HY,X

with constant K + in which I wins or p is not a legal run of the game HY,X is Borel and has open sections AY = {p ∈ (N × DK+ × N)N | (Y, p)  ∈ A}. Also, since there are no rules for the play of I in HY,X , AY really corresponds to the winning plays for I in HY,X with constant K + . By assumption, I has a winning strategy to play in AY for all Y , and so by the theorem on strategic uniformisation (see (35.32) in [24]), there is a Borel function σ : Y → σY that to each Y associates a winning strategy for I in the game HY,X with constant K + . Now let  = (δn ) be a sequence of positive reals such that for all 2KC-basic sequences of blocks (wn ) of X with K1  wn   K (where C is the basis constant of X) and sequences of vectors (un ), if for all n, wn − un  < δn , then (wn ) ∼√1+ /K (un ). We also choose sets Dn of finite (not necessarily normalised) blocks with the following properties: – for each finite d ⊆ N, the number of vectors u ∈ Dn such that supp u = d is finite, – for all blocks vectors w with K1  w  K, there is some u ∈ Dn with supp w = supp u such that w − u < δn . This is possible since the K-ball in [xi ]i∈d is totally bounded for all finite d ⊆ N. So for all 2KC-basic sequences (wn ) of blocks with K1  wn   K, there is some (un ) ∈ n Dn such that supp wn = supp un and wn − un  < δn for all n, whence (wn ) ∼√1+ /K (un ). Suppose now that Y = [yn ] is given. For each p = (n0 , u0 , m0 , . . . , ni , ui , mi ), where uj ∈ Dj for all j and n1 · · · ni I n0 II u0 , m0 u1 , m1 · · · ui , mi is a legal position in the game HY,X in which I has played according to σY , we write p < k if nj , uj , mj < k for all j  i. Notice that for all k there are only finitely many such p with p < k, so we can define 

  α(k) = max k, max σY (p)  p < k and set Ik = [k, α(k)]. Clearly, the sequence (Ik ) can be computed in a Borel fashion from Y . The Ik are not necessarily successive, but their minimal elements tend to ∞, so to prove the lemma it is enough to show that Y does not K-embed into [xn ] avoiding an infinite number of Ik including I0 . Suppose now  for a contradiction that A ⊆ N is infinite, 0 ∈ A and yi → wi is a K-embedding into [xn | n ∈ / k ∈A / Ik ]. By perturbing the embedding slightly, we can suppose that the wi are √ 1 blocks such that K  wi   K and we still have a K 1 + /K-embedding. Using the defining properties of Di , we find ui ∈ Di such that wi − ui  < δi and supp wi = supp ui for all i, whereby (ui ) ∼√1+ /K (wi ) ∼K √1+ /K (yi ), and therefore (ui ) ∼K+ (yi ). We now proceed to define natural numbers ni , mi , and ai ∈ A such that for pi = (n0 , u0 , m0 , . . . , ni , ui , mi ), we have

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(i) (ii) (iii) (iv)

163

a0 = 0 and [0, n0 [ ⊆ Ia0 , mi = ai+1 − 1, pi is a legal position in HY,X in which I has played according to σY , ]mi , ni+1 [ ⊆ Iai+1 .

Let a0 = 0 and n0 = σY (∅) = α(0), whence Ia0 = [0, α(0)] = [0, n0 ]. Find a1 such that n0 , u0 , a0 < a1 and set m0 = a1 − 1. Then p0 = (n0 , u0 , m0 ) is a legal position in HY,X in which I has played according to σY , p0 < a1 , so n1 = σY (n0 , u0 , m0 )  α(a1 ), and therefore ]m0 , n1 [ ⊆ Ia1 = [a1 , α(a1 )]. Now suppose by induction that n0 , . . . , ni and a0 , . . . , ai have been defined. Since [0, n0 [ ⊆ Ia0 and ]mj , nj +1 [ ⊆ Iaj +1 for all j < i, we have ui ∈ X[n0 , m0 ] + · · · + X[ni−1 , mi−1 ] + X[ni , ∞[ . Find some ai+1 greater than all of n0 , . . . , ni , u0 , . . . , ui , a0 , . . . , ai and let mi = ai+1 − 1. Then ui ∈ X[n0 , m0 ] + · · · + X[ni−1 , mi−1 ] + X[ni , mi ] and pi = (n0 , u0 , m0 , . . . , ni , ui , mi ) is a legal position played according to σY . Since pi < ai+1 also ni+1 = σY (n0 , u0 , m0 , . . . , ni , ui , mi )  α(ai+1 ). Thus ]mi , ni+1 [ ⊆ Iai+1 = [ai+1 , α(ai+1 )].  Now since p0 ⊆ p1 ⊆ p2 ⊆ · · · , we can let p = i pi and see that p is a run of the game in which I followed the strategy σY and II has played (ui ). Since σY is winning for I, this implies that (ui ) K+ (yi ) contradicting our assumption. 2 Lemma 3.8. Suppose X = [xn ] is a space with a basis and Y is a space such that for all constants K there are intervals I0(K) < I1(K) < I2(K) < · · · such that Y K (X, Ij(K) ). Then there are intervals J0 < J1 < J2 < · · · such that Y  (X, Jj ). Moreover, the intervals (Jj ) can be computed in (K) a Borel manner from (Ii )i,K . Proof. By induction we can construct intervals J0 < J1 < J2 < · · · such that Jn contains one (1) (n) interval from each of (Ii ), . . . , (Ii ) and if M = min Jn −1 and K = n·c(M), then max Jn > (K) max I0 + M, where c(M) is a constant such that if two subsequences of (xn ) differ in at most M terms then they are c(M) equivalent. It then follows that if A ⊆ N is infinite, then Y  [xn ]n∈/ i∈A Ji . To see this, suppose towards a contradiction that A ⊆ N is infinite and that for some integer N , Y N [xn ]n∈/ i∈A Ji .

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Pick then a ∈ A such that a  N and set M = min Ja − 1 and K = a · c(M). Define an isomorphic embedding T from 

    / Ji xn  n ∈ i∈A

into 

       (K) / xn  n ∈ Ji & n > max Ja + xn  max I0 < n  max Ja i∈A

by setting T (xn ) =

xn , xmax I (K) +n+1 , 0

if n > max Ja ; if n  M.

This is possible since max Ja > max I0(K) + M. Also, since T only changes at most M vectors from (xn ), it is a c(M) embedding. Therefore, by composing with T and using that N · c(M)  a · c(M)  K, we see that         (K) / Y K xn  n ∈ Ji & n > max Ja + xn  max I0 < n  max Ja . i∈A (K)

In particular, as almost all Ji contain an interval Il taining 0 such that

, we can find and infinite set B ⊆ N con-

   (K)   / , Ii Y K xn  n ∈ i∈B

which is a contradiction.

2

Lemma 3.9. Let E = [en ] be given and suppose that for all block subspaces Z  E and constants C there is a block subspace X  Z such that for all block subspaces Y  X, I has a winning strategy in the game HY,X with constant C. Then there is a block subspace X  E and a Borel function f : bb(X) → [N] such that for all normalised block bases Y  X, if we set Ij = [f (Y )2j , f (Y )2j +1 ], then Y  (X, Ij ). Proof. Using the hypothesis inductively together with Lemma 3.7, we can construct a sequence X0  X1  X2  · · · of block subspaces XK and corresponding Borel functions fK : bb(XK ) → [N] such that for all V  XK if Ij = [fK√(V )2j , fK (V )2j +1 ], then V K 2 (XK , Ij ). Pick by Lemma 2.2 some block X∞ of X0 that is K-equivalent with a block sequence ZK of XK for every K  1. Then for any block√ sequence Y of X∞ and any K  1 there is some block sequence V  ZK  XK such that Y is K-equivalent with V . Let (Ij ) be the intervals given

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by fK (V ) so that V K 2 (XK , Ij ). We can then in a Borel way from (Ij ) construct intervals (Jj ) such that V K 2 (ZK , Jj ) and therefore also Y K (X∞ , Jj ). This means that there are Borel functions gK : bb(X∞ ) → [N] such that for all Y  X∞ if JjK (Y ) = [gK (Y )2j , gK (Y )2j +1 ], then Y K (X∞ , JjK (Y )). Using Lemma 3.8 we can now in a Borel manner in Y define intervals LY0 < LY1 < · · · such that   Y  X∞ , LYj . Letting f : bb(X∞ ) → [N] be the Borel function corresponding to Y → (LYj ), we have our result. 2 As will be clear in Section 7 it can be useful to have a version of tightness that not only assures us that certain intervals exist, but also tells us how to obtain these. Thus, we call a basis (en ) continuously tight if there is a continuous function f : bb(en ) → [N] such that for all normalised block bases X, if we set Ij = [f (X)2j , f (X)2j +1 ], then   X  [en ], Ij , i.e., X does not embed into [en ] avoiding an infinite number of the intervals Ij . We shall now improve Lemma 3.9 to conclude continuous tightness from its hypothesis. Lemma 3.10. Let E = [en ] be given and suppose that for all block subspaces Z  E and constants C there is a block subspace X  Z such that for all block subspaces Y  X, I has a winning strategy in the game HY,X with constant C. Then there is a continuously tight block subspace X  E. Proof. We observe that E does not contain a copy of c0 . Indeed if Z is a block subspace of E spanned by a block sequence which is C-equivalent to the unit vector basis of c0 , then for any Y  X  Z, II has a winning strategy in the game HY,X with constant C 2 . We shall then use codings with inevitable subsets. So find first a block subspace Z of E such that there are inevitable, positively separated, closed subsets F0 and F1 of SZ . By Lemma 3.9, we can find a further block subspace V of Z and a Borel function g : bb(V ) → [N] such that for all Y  V , if Ij = [g(Y )2j , g(Y )2j +1 ], then Y  (V , Ij ). Define the set     

/ g (y2n+1 ) and y2n ∈ F1 ⇔ n ∈ g (y2n+1 ) . A = (yn ) ∈ bb(V )  y2n ∈ F0 ⇔ n ∈ Obviously, A is Borel and, using inevitability, one can check that any block basis of V contains a further block basis in A. Thus, by Gowers’ Determinacy Theorem, we have that for all  > 0 there is a block sequence X of V such that II has a strategy to play into A when I plays block subspaces of X. Choosing  > 0 sufficiently small, this easily implies that for some block basis X of E, there is a continuous function h : bb(X) → bb(X) × [N] that to each W  X associates a pair (Y, (In )) consisting of a block sequence Y of W and a sequence of intervals (In ) such that Y  (V , Ij ). Notice now that continuously in the sequence (Ij ), we can construct intervals (Jj ) such that Y  (X, Jj ) and hence also W  (X, Jj ). So the continuous function f : bb(X) → [N] corresponding to W → (Jj ) witnesses the continuous tightness of X. 2 We shall need the following consequence of continuous tightness in Section 7.

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Lemma 3.11. Suppose (en ) is continuously tight. Then there is a continuous function f : [N] → [N] such that for all A, B ∈ [N], if B is disjoint from an infinite number of intervals [f (A)2i , f (A)2i+1 ], then [en ]n∈A does not embed into [en ]n∈B . Proof. It is enough to notice that the function h : [N] → bb(en ) given by h(A) = (en )n∈A is continuous. So when composed with the function witnessing continuous tightness we have the result. 2 3.3. A game for minimality For L and M two block subspaces of E, define the infinite game GL,M with constant C  1 between two players as follows. In each round I chooses a subspace Ei ⊆ L spanned by a finite block sequence of L, a normalised block vector ui ∈ E0 + · · · + Ei , and an integer mi . In the first round II plays an integer n0 , and in all subsequent rounds II plays a subspace Fi spanned by a finite block sequence of M, a (not necessarily normalised) block vector vi ∈ F0 + · · · + Fi and an integer ni+1 . Moreover, we demand that ni  Ei and mi  Fi . Diagrammatically, I

n0  E 0 ⊆ L u0 ∈ E0 , m0

n1  E 1 ⊆ L u1 ∈ E0 + E1 , m1 m0  F0 ⊆ M v 0 ∈ F0 , n 1

II n0

··· m1  F1 ⊆ M · · · v 1 ∈ F0 + F1 , n 2

The outcome of the game is the pair of infinite sequences (ui ) and (vi ) and we say that II wins the game if (ui ) ∼C (vi ). Lemma 3.12. Suppose that X and Y are block subspaces of E and that player II has a winning strategy in the game HY,X with constant C. Then II has a winning strategy in the game GY,X with constant C. Proof. We shall in fact prove that II has a winning strategy in a game that is obviously harder for her to win. Namely, we shall suppose that II always plays ni = 0, which obviously puts less restrictions on the play of I. Moreover, we do not require I to play the finite-dimensional spaces Ei , which therefore also puts fewer restrictions on I in subsequent rounds. Therefore, we shall suppress all mention of Ei and ni and only require that the ui are block vectors in Y . While playing the game GY,X , II will keep track of an auxiliary play of the game HY,X in the following way. In the game GY,X we have the following play I u0 ∈ Y, m0 II

u1 ∈ Y, m1 m0  F0 ⊆ X v 0 ∈ F0

··· m1  F1 ⊆ X · · · v 1 ∈ F0 + F1

k We write each vector ui = ji=0 λij yj and may for simplicity of notation assume that ki < ki+1 . The auxiliary run of HY,X that II will keep track of is as follows, where II plays according to her winning strategy for HY,X .

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I m0 · · · m0 m1 · · · m1 ··· wk0 , pk0 wk0 +1 , pk0 +1 · · · wk1 , pk1 · · · II w 0 , p0 · · · To compute the vi and Fi in the game GY,X , II will refer to the play of HY,X and set vi =

ki 

λij wj ,

j =0

and let   Fi = X mi , max{pki−1 +1 , . . . , pki } . It is not difficult to see that mi  Fi ⊆ X, vi ∈ F0 + · · · + Fi , and that the Fi and vi only depends on u0 , . . . , ui and m0 , . . . , mi . Thus this describes a strategy for II in GY,X and it suffices to verify that it is a winning strategy. But since II follows her strategy in HY,X , we know that (wi ) ∼C (yi ) and therefore, since ui and vi are defined by the same coefficients over respectively (yi ) and (wi ), we have that (vi ) ∼C (ui ). 2 3.4. A dichotomy for minimality We are now in condition to prove the central result of this paper. Theorem 3.13 (3rd dichotomy). Let E be a Banach space with a basis (ei ). Then either E contains a minimal block subspace or a continuously tight block subspace. Proof. Suppose that E has no continuously tight block basic sequence. By Lemma 3.10, we can, modulo passing to a block subspace, suppose that for some constant C and for all block subspaces X  E there is a further block subspace Y  X such that I has no winning strategy in the game HY,X with constant C. By the determinacy of open games, this implies that for all block subspaces X  E there is a further block subspace Y  X such that II has a winning strategy in the game HY,X with constant C. A state is a pair (a, b) with a, b ∈ (D × F)<ω , where F is the set of subspaces spanned by finite block sequences and D the set of not necessarily normalised blocks, such that |a| = |b| or |a| = |b| + 1. The set S of states is countable, and corresponds to the possible positions of a game GL,M after a finite number of moves were made, restricted to elements that affect the outcome of the game from that position (i.e., mi ’s and ni ’s are forgotten). For each state s = (a, b) we will define the game GL,M (s) in a manner similar to the game GL,M depending on whether |a| = |b| or |a| = |b| + 1. To avoid excessive notation we do this via two examples: If a = (a0 , A0 , a1 , A1 ), b = (b0 , B0 , b1 , B1 ), the game GL,M (s) will start with II playing some integer n2 , then I playing (u2 , E2 , m2 ) with n2  E2 ⊆ L and u2 ∈ A0 + A1 + E2 , II playing (v2 , F2 , n3 ) with m2  F2 ⊆ M and v2 ∈ B0 + B1 + F2 , etc., and the outcome of the game will be the pair of infinite sequences (a0 , a1 , u2 , . . .) and (b0 , b1 , v2 , . . .). If a = (a0 , A0 , a1 , A1 ), b = (b0 , B0 ), the game GL,M (s) will start with I playing some integer m1 , then II playing (v1 , F1 , n2 ) with m1  F1 ⊆ M and v1 ∈ B0 + F1 , I playing (u2 , E2 , m2 )

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with n2  E2 ⊆ L and u2 ∈ A0 + A1 + E2 , etc., and the outcome of the game will be the pair of infinite sequences (a0 , a1 , u2 , . . .) and (b0 , v1 , v2 , . . .). The following lemma is well known and easily proved by a simple diagonalisation. Lemma 3.14. Let N be a countable set and let μ : bb(E) → P(N ) satisfy either V ∗ W

⇒

μ(V ) ⊆ μ(W )

V ∗ W

⇒

μ(V ) ⊇ μ(W ).

or

Then there exists a stabilising block subspace V0  E, i.e., such that μ(V ) = μ(V0 ) for any V ∗ V0 . Let now τ : bb(E) → P(S) be defined by s ∈ τ (M)

⇔

∃L  M such that player II has a winning strategy in GL,M (s).

By the asymptotic nature of the game we see that M  ∗ M ⇒ τ (M  ) ⊆ τ (M), and therefore there exists M0  E which is stabilising for τ . We then define a map ρ : bb(E) → P(S) by setting s ∈ ρ(L)

⇔

player II has a winning strategy in GL,M0 (s).

Again L ∗ L ⇒ ρ(L ) ⊇ ρ(L) and therefore there exists L0  M0 which is stabilising for ρ. Finally, the reader will easily check that ρ(L0 ) = τ (L0 ) = τ (M0 ), see, e.g., [35] or [10]. Lemma 3.15. For every M  L0 , II has a winning strategy for the game GL0 ,M . Proof. Fix M a block subspace of L0 . We begin by showing that (∅, ∅) ∈ τ (L0 ). To see this, we notice that as L0  E, there is a Y  L0 such that II has a winning strategy for HY,L0 and thus, by Lemma 3.12, also a winning strategy in GY,L0 with constant C. So (∅, ∅) ∈ τ (L0 ). We will show that for all states   (u0 , E0 , . . . , ui , Ei ), (v0 , F0 , . . . , vi , Fi ) ∈ τ (L0 ), there is an n such that for all n  E ⊆ L0 and u ∈ E0 + · · · + Ei + E, we have   (u0 , E0 , . . . , ui , Ei , u, E), (v0 , F0 , . . . , vi , Fi ) ∈ τ (L0 ). Similarly, we show that for all states   (u0 , E0 , . . . , ui+1 , Ei+1 ), (v0 , F0 , . . . , vi , Fi ) ∈ τ (L0 ) and for all m there are m  F ⊆ M and v ∈ F0 + · · · + Fi + F such that   (u0 , E0 , . . . , ui+1 , Ei+1 ), (v0 , F0 , . . . , vi , Fi , v, F ) ∈ τ (L0 ).

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Since the winning condition of GL0 ,M is closed, this clearly shows that II has a winning strategy in GL0 ,M (except for the integers m and n, τ (L0 ) is a winning quasi strategy for II). So suppose that   s = (u0 , E0 , . . . , ui , Ei ), (v0 , F0 , . . . , vi , Fi ) ∈ τ (L0 ) = ρ(L0 ), then II has a winning strategy in GL0 ,M0 (s) and hence there is an n such that for all n  E ⊆ L0 and u ∈ E0 + · · · + Ei + E, II has a winning strategy in GL0 ,M0 (s  ), where   s  = (u0 , E0 , . . . , ui , Ei , u, E), (v0 , F0 , . . . , vi , Fi ) . So s  ∈ ρ(L0 ) = τ (L0 ). Similarly, if   s = (u0 , E0 , . . . , ui+1 , Ei+1 ), (v0 , F0 , . . . , vi , Fi ) ∈ τ (L0 ) = τ (M) and m is given, then as II has a winning strategy for GL,M (s) for some L  M, there are m  F ⊆ M and v ∈ F0 + · · · + Fi + F such that II has a winning strategy in GL,M (s  ), where   s  = (u0 , E0 , . . . , ui+1 , Ei+1 ), (v0 , F0 , . . . , vi , Fi , v, F ) . So s  ∈ τ (M) = τ (L0 ).

2

Choose now Y = [yi ]  L0 such that II has a winning strategy in HY,L0 . We shall show that any block subspace M of L0 contains a C 2 -isomorphic copy of Y , which implies that Y is C 2 + -minimal for any > 0. To see this, notice that, since II has a winning strategy in HY,L0 , player I has a strategy in the game GL0 ,M to produce a sequence (ui ) that is C-equivalent with the basis (yi ). Moreover, we can ask that I plays mi = 0. Using her winning strategy for GL0 ,M , II can then respond by producing a sequence (vi ) in M such that (vi ) ∼C (ui ). So (vi ) ∼C 2 (yi ) and Y C 2 M. 2 Finally we observe that by modifying the notion of embedding in the definition of a tight basis, we obtain variations of our dichotomy theorem with a weaker form of tightness on one side and a stronger form of minimality on the other. Theorem 3.16. Every Banach space with a basis contains a block subspace E = [en ] which satisfies one of the two following properties: (1) For any [yi ]  E, there exists a sequence (Ii ) of successive intervals such that for any infinite subset A of N, the basis (yi ) does not embed into [en ]n∈/ i∈A Ii as a sequence of disjointly supported blocks , resp. as a permutation of a block sequence, resp. as a block sequence. (2) For any [yi ]  E, (en ) is equivalent to a sequence of disjointly supported blocks of [yi ], resp. (en ) is permutatively equivalent to a block sequence of [yi ], resp. (en ) is equivalent to a block sequence of [yi ]. The case of block sequences immediately implies the theorem of Pełczar [35]. The fact that the canonical basis of T ∗ is strongly asymptotically ∞ implies easily that it is tight for “embedding as a sequence of disjointly supported blocks” although T ∗ is minimal in

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the usual sense. We do not know of other examples of spaces combining one form of minimality with another form of tightness in the above list. 4. Tightness with constants and crude stabilisation of local structure We shall now consider a stronger notion of tightness, which is essentially local in nature. Let E be a space with a basis (en ). There is a particularly simple case when a sequence (Ii ) of intervals associated to a subspace Y characterises the tightness of Y in (en ). This is when for all integer constants K, Y K [en ]n∈I / K . This property has the following useful reformulations. Proposition 4.1. Let E be a space with a basis (en ). The following are equivalent: (1) For any block sequence (yn ) there are intervals I0 < I1 < I2 < · · · such that for all K, [yn ]n∈IK K [en ]n∈I / K. (2) For any space Y , there are intervals I0 < I1 < I2 < · · · such that for all K, Y K [en ]n∈I / K. (3) No space embeds uniformly into the tail subspaces of E. (4) There is no K and no subspace of E which is K-crudely finitely representable in any tail subspace of E. A basis satisfying properties (1), (2), (3), (4), as well as the space it generates, will be said to be tight with constants. Proof. The implications (1) ⇒ (2) ⇒ (3) are clear. To prove (3) ⇒ (4) assume some subspace Y of E is K-crudely finitely representable in any tail subspace of E. Without loss of generality, we may assume that Y = [yn ] is a block subspace of E. We pick a subsequence (zn ) of (yn ) in the following manner. Let z0 = y0 , and if z0 , . . . , zk−1 have been chosen, we choose zk supported far enough on the basis (en ), so that [z0 , . . . , zk−1 ] has a 2K-isomorphic copy in [en | k  n < min(supp zk )]. It follows that for any k, Z = [zn ] has an M-isomorphic copy in the tail subspace [en | n  k] for some M depending only on K and the constant of the basis (en ). To prove (4) ⇒ (1), let c(L) be a constant such that if two block sequences differ in at most L terms, then they are c(L)-equivalent. Now assume (4) holds and let (yn ) be a block sequence of (en ). Suppose also that I0 < · · · < IK−1 have been chosen. By (4) applied to l Y = [yn ]∞ n=max IK−1 +1 , we can then find m and l > max IK−1 such that [yn ]n=max IK−1 +1 does not K · c(max IK−1 + 1)-embed into [en ]∞ n=m . Let now IK = [max IK−1 + 1, l + m] and notice that, as [yn ]ln=max IK−1 +1 ⊆ [yn ]n∈IK , we have that [yn ]n∈IK does not K ·c(max IK−1 + ∞ 1)-embed into [en ]∞ n=m . Also, since (en )n=m and max IK−1 

(en )n=0

(en )∞ n=max IK−1 +1+m

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only differ in max IK−1 + 1 many terms, [yn ]n∈IK does not K-embed into max IK−1

[en ]n=0

+ [en ]∞ n=max IK−1 +1+m ,

and thus not into the subspace [en ]n∈I / K either.

2

It is worth noticing that a basis (en ), tight with constants, is necessarily continuously tight. For a simple argument shows that in order to find the intervals IK satisfying (1) above, one only needs to know a beginning of the block sequence (yn ) and hence the intervals can be found continuously in (yn ). From Proposition 4.1 we also deduce that any block basis or shrinking basic sequence in the span of a tight with constants basis is again tight with constants. There is a huge difference between the fact that no subspace of E is K-crudely finitely representable in all tails of E and then that no space is K-crudely finitely representable in all tails of E. For example, we shall see that while the former holds for Tsirelson’s space, by Dvoretzky’s theorem (see e.g. [16]), 2 is always finitely representable in any Banach space. Recall that a basis (en ) is said to be strongly asymptotically p , 1  p  +∞ [9], if there exist a constant C and a function f : N → N such that for any n, any family of n unit vectors which are disjointly supported in [ek | k  f (n)] is C-equivalent to the canonical basis of np . Proposition 4.2. Let E be a Banach space with a strongly asymptotically p basis (en ), 1  p < +∞, and not containing a copy of p . Then (en ) is tight with constants. Proof. Assume that some Banach space Y embeds with constant K in any tail subspace of E. We may assume that Y is generated by a block sequence (yn ) of E and, since any strongly asymptotically p basis is unconditional, (yn ) is unconditional. By renorming E we may assume it is 1-unconditional. By a result of W.B. Johnson [23] for any n there is a constant d(n) such that (y0 , . . . , yn ) is 2K-equivalent to a sequence of vectors in the linear span of d(n) disjointly supported unit vectors in any tail subspace of E, in particular in [ek | k  f (d(n))]. Therefore [y0 , . . . , yn ] 2KC-embeds into p . This means that Y is crudely finitely representable in p and therefore embeds into Lp , and since (yn ) is unconditional asymptotically p , that Y contains a copy of p (details of the last part of this proof may be found in [9]). 2 Corollary 4.3. Tsirelson’s space T and its convexifications T (p) , 1 < p < +∞, are tight with constants. Observe that on the contrary, the dual T ∗ of T , which is strongly asymptotically ∞ and does not contain a copy of c0 , is minimal and therefore does not contain any tight subspace. Suppose a space X is crudely finitely representable in all of its subspaces. Then there is some constant K and a subspace Y such that X is K-crudely finitely representable in all of the subspaces of Y . For if not, we would be able to construct a sequence of basic sequences (xnK ) in X such that (xnK+1 ) is a block sequence of (xnK ) and such that X is not K 2 -crudely finitely√representable in [xnK ]. By Lemma 2.2, we can then find a block sequence (yn ) of (xn0 ) that is K-equivalent with a block sequence of (xnK ) for any K and hence if X were K-crudely finitely representable in [yn ] for some K, then it would also be K 3/2 -crudely finitely representable in [xnK ], which is a contradiction.

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When a space X is K-crudely finitely representable in any of its subspaces for some K, we say that X is locally minimal. For example, by the universality properties of c0 , any space with an asymptotically ∞ basis is locally minimal. Theorem 4.4 (5th dichotomy). Let E be an infinite-dimensional Banach space with basis (en ). Then there is a block sequence (xn ) satisfying one of the following two properties, which are mutually exclusive and both possible. (1) (xn ) is tight with constants, (2) [xn ] is locally minimal. Proof. If E contains c0 , the result is trivial. So suppose not and find by the solution to the distortion problem a block sequence (yn ) and inevitable, positively separated, closed subsets F0 and F1 of the unit sphere of [yn ]. Define for each integer K  1 the set  AK = (zn )  (yn )  z2n ∈ F0 ∪ F1 and (z2n ) codes by 0’s and 1’s a block sequence (vn ) of

(z2n+1 ) such that for all N, [vn ] K [z2n+1 ]nN and moreover 1/2 < vn  < 2 . Clearly AK is analytic, so we can apply Gowers’ Determinacy Theorem to get one of two cases: (i) either there is a block sequence (xn ) and a K such that player II has a strategy to play inside (AK ) whenever I plays a block sequence of (xn ), where  will be determined later, (ii) or we can choose inductively a sequence of block sequences (xnK ) such that (xnK+1 )  (xnK ) and such that no block sequence of (xnK ) belongs to AK . n and choose now further block sequences (x ) and (h ) Consider first case (ii). Set wn = x2n n n of (wn ) such that

x0 < h0 < h1 < x1 < h2 < h3 < x4 < · · · and h2n ∈ F0 , h2n+1 ∈ F1 . We claim that (xn ) is tight with constants. If not, we can find some block sequence (un ) of (xn ) and a K such that [un ] embeds with constant K into any tail subspace of [xn ]. By passing to tails of (xn ) and of (un ), we can suppose that (xn ) is a block sequence of (xnK ), (un ) is a block sequence of (xn ) and [un ] K-embeds into all tails of [xn ]. By filling in with appropriate hi between xn and xn+1 , we can now produce a block sequence (zn ) of (xnK ) such that (z2n ) codes by 0’s and 1’s the block sequence (un ) of (z2n+1 ) with the property that for all N , [un ] K [z2n+1 ]nN . In other words, we have produced a block sequence of (xnK ) belonging to AK , which is impossible. Thus, (xn ) is tight with constants. Consider now case (i) instead and let II play according to her strategy. We suppose that  is chosen sufficiently small so that δi < dist(F0 , F1 )/3 and if two block sequences are -close then they are 2-equivalent. Let (yn ) ∈ (AK ) be the response by II to the sequence (xn ) played by I and let (zn ) ∈ AK be such that zn − yn  < δn for all n. Then (z2n ) codes by 0’s and 1’s a block sequence (vn ) of (z2n+1 ). Let (un ) be the block sequence of (y2n+1 ) constructed in the same way as (vn ) is constructed over (z2n+1 ). We claim that (un ) is 4K-crudely finitely representable in any block subspace of [xn ].

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For this, let [u0 , . . . , um ] be given and suppose that (fn ) is any block subspace of (xn ). Find a large k such that (z0 , z2 , . . . , z2k ) codes the block sequence (v0 , . . . , vm ) of (z1 , . . . , z2k+1 ) and let l be large enough so that when I has played x0 , . . . , xl then II has played y0 , . . . , y2k+1 . Consider now the game in which player I plays x0 , x1 , . . . , xl , fl+1 , fl+2 , . . . . Then, following the strategy, II will play a block sequence y0 , . . . , y2k+1 , g2k+2 , g2k+3 , . . . ∈ (AK ) . So let (hn ) ∈ AK be such that hn − yn  < δn for all n  2k + 1 and hn − gn  < δn for all n  2k + 2. For n  k, we have, as h2n − z2n  < 2δn < 23 dist(F0 , F1 ), that h2n ∈ Fi ⇔ ∞ z2n ∈ Fi . Also, (h2n+1 )kn=0 and (y2n+1 )kn=0 are 2-equivalent and (h2n+1 )∞ n=k+1 and (g2n+1 )n=k+1 are 2-equivalent, so (h2n ) will code a block sequence (wn ) of (h2n+1 ) such that (w0 , . . . , wm ) is 2-equivalent to (u0 , . . . , um ). Moreover, since (hn ) ∈ AK , [wn ] will K-embed into every tail subspace of [h2n+1 ], and hence 2K-embed into every tail subspace of [g2n+1 ]. Therefore, since (g2n+1 ) is block sequence of (fn ), [u0 , . . . , um ] will 4K-embed into [fn ], which proves the claim. It follows that [un ] is locally minimal, which proves the theorem. 2 Local minimality can be reformulated in a way that makes the relation to local theory clearer. For this, let Fn be the metric space of all n-dimensional Banach spaces up to isometry equipped with the Banach–Mazur metric       d(X, Y ) = inf log T  · T −1   T : X → Y is an isomorphism . Then for every Banach space X, the set of n-dimensional Y that are almost isometrically embeddable into X form a closed subset (X)n of Fn . It is well known that this set (X)n does not always stabilise, i.e., there is not necessarily a subspace Y ⊆ X such that for all further subspaces Z ⊆ Y , (Z)n = (Y )n . However, if instead X comes equipped with a basis and for all block subspaces Y we let {Y }n be the set of all n-dimensional spaces that are almost isometrically embeddable into all tail subspaces of Y , then one can easily stabilise {Y }n on subspaces. Such considerations are for example the basis for [30]. Theorem 4.4 gives a dichotomy for when one can stabilise the set (X)n in a certain way, which we could call crude. Namely, X is locally minimal if and only if there is some constant K such that for all subspaces Y of X and all n, dH ((X)n , (Y )n )  K, where dH is the Hausdorff distance. So by Theorem 4.4, the local structure stabilises crudely on a subspace if and only if a space is not saturated by basic sequences tight with constants. Often it is useful to have a bit more than local minimality. So we say that a basis (en ) is locally block minimal if it is K-crudely block finitely representable in all of its block bases for some K. As with crude finite representability we see that there then must be a constant K and a block (yn ) such that (en ) is K-crudely block finitely representable in all block subspaces of (yn ). We now have the following version of Theorem 4.4 for finite block representability. Theorem 4.5. Let (en ) be a Schauder basis. Then (en ) has a block basis (xn ) with one of the following two properties, which are mutually exclusive and both possible.

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(1) For all block bases (yn ) of (xn ) there are intervals I1 < I2 < I3 < · · · such that (yn )n∈IK is not K-equivalent to a block sequence of (xn )n∈I / K, (2) (xn ) is locally block minimal. Finally we note that there exist tight spaces which do not admit subspaces which are tight with constants: Example 4.6. There exists a reflexive, tight, locally block minimal Banach space. Proof. E. Odell and T. Schlumprecht [33] have built a reflexive space OS with a basis such that every monotone basis is block finitely representable in any block subspace of OS. It is in particular locally block minimal and therefore contains no basic sequence which is tight with constants. We do not know whether the space OS is tight. Instead, we notice that since the summing basis of c0 is block finitely representable in any block subspace of OS, OS cannot contain an unconditional block sequence. By Gowers’ 1st dichotomy it follows that some block subspace of OS is HI, and by the 3rd dichotomy (Theorem 3.13) and the fact that HI spaces do not contain minimal subspaces, that some further block subspace is tight, which completes the proof. 2 It is unknown whether there is an unconditional example with the above property. There exists an unconditional version of OS [32], but it is unclear whether it has no minimal subspaces. However, the dual of a space constructed by Gowers in [17] can be shown to be both tight and locally minimal. Example 4.7. (See [14].) There exists a space with an unconditional basis which is tight and locally minimal. 5. Local block minimality, asymptotic structure and a dichotomy for containing c0 or p Recall that a basis (en ) is said to be asymptotically p (in the sense of Tsirelson’s space) if there is a constant C such that for all normalised block sequences n < x1 < · · · < xn , (xi )ni=1 is C-equivalent with the standard unit vector basis of np . When (xn ) is asymptotically p and some block subspace [yn ] of [xn ] is K-crudely block finitely representable in all tail subsequences of (xn ), then it is clear that (yn ) must actually be equivalent with the unit vector basis of p , or c0 for p = ∞. So this shows that for asymptotically p bases (xn ), either [xn ] contains an isomorphic copy of p or c0 or (xn ) itself satisfies condition (1) of Theorem 4.5. This is the counterpart of Proposition 4.2 for block sequences. As an example, we mention that, since T ∗ does not contain c0 , but has a strongly asymptotically ∞ basis, it thus satisfies (1). This small observation indicates that one can characterise when p or c0 embeds into a Banach space by characterising when a space contains an asymptotic p space. We first prove a dichotomy for having an asymptotic p subspace, for the proof of which we need the following lemma. Lemma 5.1. Suppose (en ) is a basic sequence such that for some C and all n and normalised block sequences n < y1 < y2 < · · · < y2n , we have (y2i−1 )ni=1 ∼C (y2i )ni=1 . Then (en ) has an asymptotic p subsequence for some 1  p  ∞.

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Proof. By the theorem of Brunel and Sucheston [5], we can, by passing to a subsequence of (en ), suppose that (en ) generates a spreading model, i.e., we can assume that for all integers n < l1 < l2 < · · · < ln and n < k1 < k2 < · · · < kn we have (el1 , . . . , eln ) ∼1+ 1 (ek1 , . . . , ekn ). n

Now suppose that en < y1 < y2 < · · · < yn and en < z1 < · · · < zn are normalised block sequences of (e2i ). Then there are n < l1 < l2 < · · · < ln and n < k1 < k2 < · · · < kn such that e n < y 1 < e l1 < y 2 < e l2 < · · · < y n < e ln and en < z1 < ek1 < z2 < ek2 < · · · < zn < ekn , so (yi ) ∼C (eli ) ∼1+ 1 (eki ) ∼C (zi ) and (yi ) ∼2C 2 (zi ). Thus, asymptotically all finite norn

malised block sequences are 2C 2 -equivalent. By Krivine’s theorem [25], there is some p that is block finitely representable in (e2i ) and hence asymptotically all finite normalised block sequences are equivalent to np of the correct dimension and (e2i ) is asymptotic p . 2 Theorem 5.2. Suppose X is a Banach space with a basis. Then X has a block subspace W , which is either asymptotic p , for some 1  p  +∞, or such that   ∀M ∃n ∀U1 , . . . , U2n ⊆ W ∃ui ∈ SUi u1 < · · · < u2n & (u2i−1 )ni=1 M (u2i )ni=1 . Proof. Assume first that for some M and V ⊆ X we have ∀n ∀Y ⊆ V ∃Z ⊆ Y ∀z1 < · · · < z2n ∈ SZ (z2i−1 )ni=1 ∼M (z2i )ni=1 . Then we can inductively define V ⊇ Z1 ⊇ Z2 ⊇ Z3 ⊇ · · · such that for each n, ∀z1 < · · · < z2n ∈ SZn (z2i−1 )ni=1 ∼M (z2i )ni=1 . Diagonalising over this sequence, we can find a block subspace W = [wn ] such that for all m  n, wm ∈ Zn . Therefore, if n < z1 < · · · < z2n is a sequence of normalised blocks of (wn ), then (z2i−1 )ni=1 ∼M (z2i )ni=1 . By Lemma 5.1, it follows that W has an asymptotic p subspace. So suppose on the contrary that ∀M ∀V ⊆ X ∃n ∃Y ⊆ V ∀Z ⊆ Y ∃z1 < · · · < z2n ∈ SZ (z2i−1 )ni=1 M (z2i )ni=1 . Now find some small  > 0 such that two normalised block sequences of X that are -close are √ 2-equivalent. Applying the determinacy theorem of Gowers to  and the sets 

A(M, n) = (zi )  (z2i−1 )ni=1 M (z2i )ni=1 ,

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we have that ∀M ∀V ⊆ X ∃n ∃Z ⊆ V II has a strategy to play normalised z1 < · · · < z2n such that (z2i−1 )ni=1  M (z2i )ni=1 2

when I is restricted to playing blocks of Z. Using this we can inductively define a sequence X ⊇ W1 ⊇ W2 ⊇ W3 ⊇ · · · such that for all M there is an n = n(2M) such that II has a strategy to play normalised z1 < · · · < z2n such that (z2i−1 )ni=1 M (z2i )ni=1 whenever I is restricted to playing blocks of WM . Diagonalising over this sequence, we find some W ⊆ X such that for all M, W ⊆∗ WM . So for all M there is n such that II has a strategy to play normalised z1 < · · · < z2n such that (z2i−1 )ni=1 M (z2i )ni=1 whenever I is restricted to playing blocks of W . Letting player I play a segment of the block basis of Ui until II plays a vector, we easily see that whenever U1 , . . . , U2n ⊆ W , there are zi ∈ SUi such that z1 < · · · < z2n and (z2i−1 )ni=1 M (z2i )ni=1 . This finishes the proof. 2 Using this, we can now prove the main result of this section. Theorem 5.3 (The c0 and p dichotomy). Suppose X is a Banach space not containing a copy of c0 nor of p , 1  p < ∞. Then X has a subspace Y with a basis satisfying one of the following properties. (i) ∀M ∃n ∀U1 , . . . , U2n ⊆ Y ∃ui ∈ SUi u1 < · · · < u2n & (u2i−1 )ni=1 M (u2i )ni=1 . (ii) For all block bases (zn ) of Y = [yn ] there are intervals I1 < I2 < I3 < · · · such that (zn )n∈IK is not K-equivalent to a block sequence of (yn )n∈I / K. Notice that both (i) and (ii) trivially imply that Y cannot contain a copy of c0 or p , since (i) implies some lack of homogeneity and (ii) some lack of minimality. However, we do not know if there are any spaces satisfying both (i) and (ii) or if, on the contrary, these two properties are incompatible. Proof. If X has no block subspace satisfying (i), then it must have an asymptotically p block subspace Y for some 1  p  ∞. Let C be the constant of asymptoticity. Suppose now that Z = [zn ] is a further block subspace. Since Z is not isomorphic to p , this means that for all M−L−1 and hence not L and K, there is some M such that (zn )M n=L is not KC-equivalent to p ∞ K-equivalent with a normalised block sequence of (yn )n=M either. So if I1 < I2 < · · · < IK−1 have been defined, to define IK , we let N = max IK−1 + 1 and find M such that (zn )M n=2N −1 is M not K-equivalent with a normalised block sequence of (yn )∞ . It follows that (z ) n n=N is not n=M K-equivalent with a normalised block sequence of (y1 , y2 , . . . , yN −1 , yM+1 , yM+2 , . . .). Letting IK = [N, M] we have the result. 2 We should mention that G. Androulakis, N. Kalton and Tcaciuc [1] have extended Tcaciuc’s dichotomy from [41] to a dichotomy characterising containment of p and c0 . The result above implies theirs and moreover provides additional information.

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6. Tightness by range and subsequential minimality Theorem 1.1 shows that if one allows oneself to pass to a basis for a subspace, one can find a basis in which there is a close connection between subspaces spanned by block bases and subspaces spanned by subsequences. Thus, for example, if the basis is tight there can be no space embedding into all the subspaces spanned by subsequences of the basis. On the other hand, any block basis in Tsirelson’s space T is equivalent to a subsequence of the basis, and actually every subspace of a block subspace [xn ] in T contains an isomorphic copy of a subsequence of (xn ). In fact, this phenomenon has a deeper explanation and we shall now proceed to show that the connection between block sequences and subsequences can be made even closer. Lemma 6.1. If (en ) is a basis for a space not containing c0 , then for all finite intervals (In ) such that min In − −−− → ∞ and all subspaces Y , there is a further subspace Z such that n→∞ PIk |Z  − −−→ 0. k→∞ Proof. By a standard perturbation argument, we can suppose that Y is generated by a normalised block basis (yn ). Let K be the basis constant of (en ). As min In − −−− → ∞ and each In is finite, n→∞ we can choose a subsequence (vn ) of (yn ) such that for all k the interval Ik intersects the range of at most one vector vm from (vn ). Now, since c0 does not embed into [en ], no tail sequence of (vn ) can satisfy an upper c0 estimate. This implies that for all N and δ > 0 there is a normalised vector 

z=

N 

ηi v i ,

i=N

where |ηi | < δ. Using this, we now construct a normalised block sequence (zn ) of (vn ) such that there are m(0) < m(1) < · · · and αi with zn =

m(n+1)−1 

αi vi

i=m(n)

and |αi | < n1 whenever

m(n)  i < m(n + 1). Now suppose u = j λj zj and k are given. Then there is at most one vector zn whose range intersect the interval Ik . Also, there is at most one vector vp from the support of zn whose range intersect Ik . Therefore,       PI (u) = PI (λn zn ) = PI (λn αp vp ) k k k  2Kλn αp vp   |λn | · 2

4K 2 2K  u. n n

It follows that PIk |[zl ]   4K nk , where nk is such that Ik intersects the range of znk (or nk = k if Ik intersects the range of no zn ). Since min Ik − −−→ ∞ and (zn ) is a block basis, nk → ∞ when k→∞ k → ∞, and hence PIk |[zl ]  − − − → 0. 2 k→∞

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Our next result should be contrasted with the construction by Pełczy´nski [36] of a basis (fi ) such that every basis is equivalent with a subsequence of it, and hence such that every space contains an isomorphic copy of a subsequence. We shall see that for certain spaces E such constructions cannot be done relative to the subspaces of E provided that we demand that (fn ) lies in E too. Recall that two Banach spaces are said to be incomparable if neither of them embeds into the other. Proposition 6.2. Suppose that (en ) is a basis such that any two block subspaces with disjoint ranges are incomparable. Suppose also that (fn ) is either a block basis or a shrinking basic sequence in [en ]. Then [en ] is saturated with subspaces Z such that no subsequence of (fn ) embeds into Z. Proof. Let Y be an arbitrary subspace of [en ]. Suppose first that (fn ) is a normalised shrinking basic sequence. Then, by taking a perturbation of (fn ), we can suppose that each fn is a finite block vector of (ei ) and, moreover, that min range(fn ) → ∞. Let In = range(fn ). Fix an infinite set N ⊆ N. Then for all infinite subsets A ⊆ N there is an infinite subset B ⊆ A such that (fn )n∈B is a block sequence and hence, since block subspaces of (en ) with disjoint ranges are incomparable, [fn ]n∈B  [en ]n∈/ i∈B Ii , and so also [fn ]n∈N  [en ]n∈/ i∈A Ii . Applying Lemma 3.2 to X = [fn ]n∈N and (In )n∈N , this implies that for all embeddings T : [fn ]n∈N → [en ]n∈N , we have lim infn∈N PIk T  > 0. So find by Lemma 6.1 a subspace −−→ 0. Then no subsequence of (fn )∈N embeds into Z. Z ⊆ Y such that PIk |Z  − k→∞ The argument in the case (fn ) is a block basis is similar. We set In = range fn and repeat the argument above. 2 We notice that in the above proof we actually have a measure for how “flat” a subspace Z of [en ] needs to be in order that the subsequences of (fn ) cannot embed into Z. Namely, it suffices −−→ 0. that PIk |Z  − k→∞ We should also mention that, by similar but simpler arguments, one can show that if (en ) is a basis such that any two disjoint subsequences span incomparable spaces, then some subspace of [en ] fails to contain any isomorphic copy of a subsequence of (en ). The assumption in Proposition 6.2 that block subspaces with disjoint ranges are incomparable is easily seen to be equivalent to the following property of a basis (en ), that we call tight by range. If (yn ) is a block sequence of (en ) and A ⊆ N is infinite, then      / [yn ]n∈N  en  n ∈ range yi . i∈A

Thus, (en ) is tight by range if it is tight and for all block sequences (yn ) of (en ) the corresponding sequence of intervals Ii is given by Ii = range yi . This property is also weaker than disjointly supported subspaces being incomparable, which we shall call tight by support. It is trivial to see that a basis, tight by range, is continuously tight. We say that a basic sequence (en ) is subsequentially minimal if any subspace of [en ] contains an isomorphic copy of a subsequence of (en ). It is clearly a weak form of minimality. In [26] the authors study another notion in the context of certain partly modified mixed Tsirelson spaces that they also call subsequential minimality. According to their definition, a basis (en ) is subsequentially minimal if any block basis has a further block basis equivalent to a subsequence of (en ). However, in all their examples the basis (en ) is weakly null and it is easily

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seen that whenever this is the case the two definitions agree. They also define (en ) to be strongly non-subsequentially minimal if any block basis contains a further block basis that has no further block basis equivalent to a subsequence of (en ). By Proposition 6.2, this is seen to be weaker than tightness by range. We shall now proceed to show a dichotomy between tightness by range and subsequential minimality. Theorem 6.3 (4th dichotomy). Let E be a Banach space with a basis (en ). Then there exists a block sequence (xn ) of (en ) with one of the following properties, which are mutually exclusive and both possible: (1) Any two block subspaces of [xn ] with disjoint ranges are incomparable. (2) The basic sequence (xn ) is subsequentially minimal. Arguably Theorem 6.3 is not a dichotomy in Gowers’ sense, since property (2) is not hereditary: for example the universal basis of Pełczy´nski [36] satisfies (2) while admitting subsequences with property (1). However, it follows obviously from Theorem 6.3 that any basis (en ) either has a block basis such that any two block subspaces with disjoint ranges are incomparable or has a block basis (xn ) that is hereditarily subsequentially minimal, i.e., such that any block basis has a further block basis that is subsequentially minimal. Furthermore, by an easy improvement of our proof or directly by Gowers’ second dichotomy, if the first case of Theorem 6.3 fails, then one can also suppose that [xn ] is quasi minimal. We shall call a basis (xn ) sequentially minimal if it is both hereditarily subsequentially minimal and quasi minimal. This is equivalent to any block basis of (xn ) having a further block basis (yn ) such that every subspace of [xn ] contains an equivalent copy of a subsequence of (yn ). We may therefore see Theorem 6.3 as providing a dichotomy between tightness by range and sequential minimality. Before giving the proof of Theorem 6.3, we first need to state an easy consequence of the definition of Gowers’ game. Lemma 6.4. Let E be a space with a basis and assume II has a winning strategy in Gowers’ game in E to play in some set B. Then there is a non-empty tree T of finite block sequences such that [T ] ⊆ B and for all (y0 , . . . , ym ) ∈ T and all block sequences (zn ) there is a block ym+1 of (zn ) such that (y0 , . . . , ym , ym+1 ) ∈ T . Proof. Suppose σ is the strategy for II. We define a pruned tree T of finite block bases (y0 , . . . , ym ) and a function ψ associating to each (y0 , . . . , ym ) ∈ T a sequence (z0 , . . . , zk ) such that for some k0 < · · · < km = k, I z0 · · · zk0 zk0 +1 · · · zk1 · · · zkm−1 +1 · · · zkm II y0 y1 · · · ym has been played according to σ . • The empty sequence ∅ is in T and ψ(∅) = ∅. • If (y0 , . . . , ym ) ∈ T and ψ(y0 , . . . , ym ) = (z0 , . . . , zk ),

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then we let (y0 , . . . , ym , ym+1 ) ∈ T if there are some zk < zk+1 < · · · < zl and k0 < · · · < km = k such that zk0 +1 · · · zk1 · · · zkm +1 · · · zl I z0 · · · zk0 II y0 y1 · · · ym+1 has been played according to σ and in this case we let ψ(y0 , . . . , ym , ym+1 ) = (z0 , . . . , zk , zk+1 , . . . , zl ) be some such sequence. Now, if (y0 , y1 , y2 , . . .) is such that (y0 , . . . , ym ) ∈ T for all m, then ψ(∅) ⊆ ψ(y0 ) ⊆ ψ(y  0 , y1 ) ⊆ · · · and (yi ) is the play of II according to the strategy σ in response to (zi ) = n ψ(y0 , . . . , yn ) being played by I. So [T ] ⊆ B. It also follows by the construction that for each (y0 , . . . , ym ) ∈ T and block sequence (zi ) there is a block ym+1 of (zi ) such that (y0 , . . . , ym , ym+1 ) ∈ T . 2 We now pass to the proof of Theorem 6.3. Proof. If E contains c0 the theorem is trivial. So suppose not. By the solution to the distortion problem and by passing to a subspace, we can suppose there are two positively separated inevitable closed subsets F0 and F1 of the unit sphere of E, i.e., such that dist(F0 , F1 ) > 0 and every block basis has block vectors belonging to both F0 and F1 . Suppose that (en ) has no block sequence satisfying (1). Then for all block sequences (xn ) there are further block sequences (yn ) and (zn ) with disjoint ranges such that [yn ]  [zn ]. We claim that there is a block sequence (fn ) and a constant K such that for all block sequences (xn ) of (fn ) there are further block sequences (yn ) and (yn ) with disjoint ranges such that [yn ] K [zn ]. If not, we can construct a sequence of block sequences (fnK ) such that (fnK+1 ) is a block of (fnK ) and such that any two block sequences of (fnK ) with disjoint ranges are K 2 -incomparable. √ By Lemma 2.2, we then find a block sequence (gn ) of (en ) that is K-equivalent with a block sequence of (fnK ) for every K  1. Find now block subspaces (yn ) and (zn ) of (gn ) with disjoint √ ranges and a K such that [yn ] √K [zn ]. Then (gn ) is K-equivalent with a block sequence of (fnK ) and hence we can find K 3/2 -comparable block subspaces of (fnK ) with disjoint ranges, contradicting our assumption. So suppose (fn ) and K are chosen as in the claim. Then for all block sequences (xn ) of (fn ) we can find an infinite set B ⊆ N and a block sequence (yn ) of (xn ) such that [yn ]n∈B K-embeds into [yn ]n∈B / . We claim that any block basis of (fn ) has a further block basis in the following set of normalised block bases of (fn ): 

A = (yn )  ∀n y2n ∈ F0 ∪ F1 & ∃∞ n y2n ∈ F0 & [y2n+1 ]y2n ∈F0 K [y2n+1 ]y2n ∈F1 . To see this, suppose that (xn ) is a block sequence of (fn ) and let (zn ) be a block sequence of (xn ) such that z3n ∈ F0 and z3n+1 ∈ F1 . We can now find an infinite set B ⊆ N and a block sequence (vn ) of (z3n+2 ) such that [vn ]n∈B K [vn ]n∈B / . Let now y2n+1 = vn and notice that we

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can choose y2n = zi ∈ F0 for n ∈ B and y2n = zi ∈ F1 for n ∈ / B such that y0 < y1 < y2 < · · · . Then (yn ) ∈ A. Choose now a sequence  = (δn ) of positive reals, δn < dist(F0 , F1 )/3, such that if (xn ) and (yn ) are block bases of (en ) with xn − yn  < δn , then (xn ) ∼2 (yn ). Since A is clearly analytic, it follows by Gowers’ determinacy theorem that for some block basis (xn ) of (fn ), II has a winning strategy to play in A whenever I plays a block basis of (xn ). We now show that some block basis (vn ) of (xn ) is such that any subspace of [vn ] contains a sequence 2K-equivalent to a subsequence of (vn ), which will give us case (2). Pick first by Lemma 6.4 a non-empty tree T of finite block sequences of (xn ) such that [T ] ⊆ A and for all (u0 , . . . , um ) ∈ T and all block sequences (zn ) there is a block um+1 of (zn ) such that (u0 , . . . , um , um+1 ) ∈ T . Since T is countable, we can construct inductively a block sequence (vn ) of (xn ) such that for all (u0 , . . . , um ) ∈ T there is some vn with (u0 , . . . , um , vn ) ∈ T . We claim that (vn ) works. For if (zn ) is any block sequence of (vn ), we construct inductively a sequence (un ) ∈ A as follows. Using inductively the extension property of T , we can construct an infinite block sequence (h0n ) of (zn ) that belongs to [T ]. Since [T ] ⊆ A , there is a shortest initial segment (u0 , . . . , u2k0 ) ∈ T of (h0n ) such that d(u2k0 , F0 ) < δ2k0 . Pick now a term u2k0 +1 from (vn ) such that (u0 , . . . , u2k0 , u2k0 +1 ) ∈ T . Again, using the extension property of T , there is an infinite block sequence (h1n ) of (zn ) such that   (u0 , . . . , u2k0 , u2k0 +1 ) h1n n ∈ [T ]. Also, as [T ] ⊆ A , there is a shortest initial segment (u0 , . . . , u2k0 , u2k0 +1 , . . . , u2k1 ) ∈ T of   (u0 , . . . , u2k0 , u2k0 +1 ) h1n n that properly extends (u0 , . . . , u2k0 , u2k0 +1 ) and such that d(u2k1 , F0 ) < δ2k1 . We then pick a term u2k1 +1 of (vn ) such that (u0 , . . . , u2k1 , u2k1 +1 ) ∈ T . We continue in the same fashion. At infinity, we then have a block sequence (un ) ∈ A and integers k0 < k1 < · · · such that d(u2n , F0 ) < δ2n if and only if n = ki for some i and such that for every i, u2ki +1 is a term of (vn ). Let now (wn ) ∈ A be such that wn − un  < δn . Then, as δn < dist(F0 , F1 )/3, we have that w2n ∈ F0 if and only if n = ki for some i and w2n ∈ F1 otherwise. Moreover, as (wn ) ∈ A, [w2ki +1 ]i∈N = [w2n+1 ]w2n ∈F0 K [w2n+1 ]w2n ∈F1 = [w2n+1 ]n=ki . So by the choice of δn we have [u2ki +1 ]i∈N 2 [w2ki +1 ]i∈N K [w2n+1 ]n=ki 2 [u2n+1 ]n=ki . Since [u2n+1 ]n=ki is a subspace of [zn ] and (u2ki +1 ) a subsequence of (vn ) this finishes the proof. 2 If, for some constant C, all subspaces of [en ] contain a C-isomorphic copy of a subsequence of (en ), we say that (en ) is subsequentially C-minimal. Our proof shows that condition (2) in

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Theorem 6.3 may be improved to “For some constant C the basic sequence (xn ) is subsequentially C-minimal”. We notice that if (xn ) is hereditarily subsequentially minimal, then there is some C and a block sequence (vn ) of (xn ) such that (vn ) is hereditarily subsequentially C-minimal with the obvious definition. To see this, we first notice that by Proposition 6.2, (xn ) can have no block bases (yn ) such that further block subspaces with disjoint ranges are incomparable. So, by the proof of Theorem 6.3, for any block base (yn ) there is a constant C and a further block basis (zn ) which is subsequentially C-minimal. A simple diagonalisation using Lemma 2.2 now shows that by passing to a block (vn ) the C can be made uniform. Recall that Gowers also proved that a quasi minimal space must contain a further subspace which is C-quasi minimal [20]. We also indicate a variation on Theorem 6.3, relating the Casazza property to a slightly stronger form of sequential minimality. This answers the original problem of Gowers left open in [20], which was mentioned in Section 1. This variation is probably of less interest than Theorem 6.3 because the Casazza property does not seem to imply tightness and also because the stronger form of sequential minimality may look somewhat artificial (although it is satisfied by Tsirelson’s space and is reminiscent of Schlumprecht’s notion of Class 1 space [40]). We say that two block sequences (xn ) and (yn ) alternate if either x0 < y0 < x1 < y1 < · · · or y0 < x 0 < y 1 < x 1 < · · · . Proposition 6.5. Let E be a Banach space with a basis (en ). Then there exists a block sequence (xn ) with one of the following properties, which are exclusive and both possible: (1) [xn ] has the Casazza property, i.e., no alternating block sequences in [xn ] are equivalent. (2) There exists a family B of block sequences saturating [xn ] and such that any two block sequences in B have subsequences which alternate and are equivalent. In particular, in case (2), E contains a block subspace U = [un ] such that for every block sequence of U , there is a further block sequence equivalent to, and alternating with, a subsequence of (un ). Proof. If (en ) does not have a block sequence satisfying (1), then any block sequence of (en ) has a further block sequence in A = {(yn ) | (y2n ) ∼ (y2n+1 )}. Let  be small enough so that A = A. By Gowers’ theorem, let (xn ) be some block sequence of (en ) so that II has a winning strategy to play in A whenever plays a block sequence of (xn ). Let T be the associated tree given by Lemma 6.4. By construction, for any block sequence (zn ) of (xn ), we may find a further block sequence (vn ) such that for any (y0 , . . . , ym ) ∈ T , there exists some vn with (y0 , . . . , ym , vn ) ∈ T . We set f ((zn )) = (vn ) and B = {f ((zn )) | (zn )  (xn )}. Given (vn ) and (wn ) in B, it is then clear that we may find subsequences (vn ) and (wn ) so that (v0 , w0 , v1 , w1 , . . .) ∈ T , and therefore (vn ) and (wn ) are equivalent. 2 We may also observe that there is no apparent relation between tightness by range and tightness with constants. Indeed Tsirelson’s space is tight with constants and sequentially minimal. Similarly, Example 4.7 is tight by support and therefore by range, but is locally minimal. Using the techniques of [3], one can construct a space combining the two forms of tightness. Example 6.6. (See [14].) There exists a space with a basis which is tight with constants and tight by range.

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Finally, if a space X is locally minimal and equipped with a basis which is tight by support and therefore unconditional (such as Example 4.7), then the reader will easily check the following. The canonical basis of X ⊕ X is tight (for a block subspace Y = [yn ] of X ⊕ X use the sequence of intervals associated to the ranges of yn with respect to the canonical 2-dimensional decomposition of X ⊕ X), but neither tight by range nor with constants. However, a more interesting question remains open: does there exist a tight space which does not contain a basic sequence which is tight by range or with constants? There is a natural strengthening of sequential minimality that has been considered in the literature, namely, the blocking principle (also known as the shift property in [7]) due to Casazza, Johnson, and Tzafriri [6]. It is known that for a normalized unconditional basis (en ) the following properties are equivalent (see, e.g., [12]). (1) Any block sequence (xn ) spans a complemented subspace of [en ]. (2) For any block sequence (xn ), (x2n ) ∼ (x2n+1 ). (3) For any block sequence (xn ) and integers kn ∈ supp xn , (xn ) ∼ (ekn ). Moreover, any of the above properties necessarily hold uniformly. We say that (en ) satisfies the blocking principle if the above properties hold for (en ). The following proposition can be proved along the lines of the proof of the minimality of T ∗ in [6] (Theorem 14). Proposition 6.7. Let (en ) be an unconditional basis satisfying the blocking principle and spanning a locally minimal space. Then (en ) spans a minimal space. Thus, by the 5th dichotomy (Theorem 4.4), we have Corollary 6.8. Let (en ) be an unconditional basis satisfying the blocking principle. Then there is a subsequence (fn ) of (en ) such that either [fn ] is minimal or (fn ) is tight with constants. 7. Chains and strong antichains The results in this section are in response to a question of Gowers from his fundamental study [20] and concern what types of quasi orders can be realised as the set of (infinitedimensional) subspaces of a fixed Banach space under the relation of isomorphic embeddability. Problem 7.1. (See Problem 7.9. in [20].) Given a Banach space X, let P(X) be the set of all equivalence classes of subspaces of X, partially ordered by isomorphic embeddability. For which posets P does there exist a Banach space X such that every subspace Y of X contains a further subspace Z with P(Z) = P ? Gowers noticed himself that by a simple diagonalisation argument any such poset P(X) must either have a minimal element, corresponding to a minimal space, or be uncountable. We shall now use our notion of tightness to show how to attack this problem in a uniform way and improve on several previous results. Suppose X is a separable Banach space and let SB(X) denote the set of all closed linear subspaces of X. We equip SB(X) with the so-called Effros–Borel structure, which is the σ algebra generated by sets on the form  {Y ∈ SB(X)  Y ∩ U = ∅},

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where U is an open subset of X. In this way, SB(X) becomes a standard Borel space, i.e., isomorphic as a measurable space to the real line equipped with its Borel algebra. We refer to the measurable subsets of SB(X) as Borel sets. Let also SB∞ (X) be the subset of SB(X) consisting of all infinite-dimensional subspaces of X. Then SB∞ (X) is a Borel subset of SB(X) and hence a standard Borel space in its own right. Definition 7.2. Suppose that X is a separable Banach space and E is an analytic equivalence relation on a Polish space Z. We say that X has an E-antichain, if there is a Borel function f : Z → SB(X) such that for x, y ∈ Z (1) if xEy, then f (x) and f (y) are biembeddable, (2) if xE / y, then f (x) and f (y) are incomparable. We say that X has a strong E-antichain if there is a Borel function f : Z → SB(X) such that for x, y ∈ Z (1) if xEy, then f (x) and f (y) are isomorphic, / y, then f (x) and f (y) are incomparable. (2) if xE For example, if =R is the equivalence relation of identity on R, then =R -antichains and strong =R -antichains simply correspond to a perfect antichain in the usual sense, i.e., an uncountable Borel set of pairwise incomparable subspaces. Also, having a strong E-antichain implies, in particular, that E Borel reduces to the isomorphism relation between the subspaces of X. The main result of [11] reformulated in this language states that if EΣ 1 denotes the complete 1 analytic equivalence relation, then C[0, 1] has a strong EΣ 1 -antichain. 1 We will now prove a result that simultaneously improves on two results due respectively to the first and the second author. In [12], the authors proved that a Banach space not containing a minimal space must contain a perfect set of non-isomorphic subspaces. This result was improved by Rosendal in [37], in which it was shown that if a space does not contain a minimal subspace it must contain a perfect set of pairwise incomparable spaces. And Ferenczi proved in [10] that if X is a separable space without minimal subspaces, then E0 Borel reduces to the isomorphism relation between the subspaces of X. Recall that E0 is the equivalence relation defined on 2N by xE0 y if and only if ∃m ∀n  m xn = yn . Theorem 7.3. Let X be a separable Banach space. Then X either contains a minimal subspace or has a strong E0 -antichain. Proof. Suppose X has no minimal subspace. By Theorem 3.13 and Lemma 3.11, we can find a basic sequence (en ) in X and a continuous function f : [N] → [N] such that for all A, B ∈ [N], if B is disjoint from an infinite number of intervals [f (A)2i , f (A)2i+1 ], then [en ]n∈A does not embed into [en ]n∈B . We claim that there is a continuous function h : 2N → [N] such that (1) if xE0 y, then |h(x) \ h(y)| = |h(y) \ h(x)| < ∞, / 0 y, then [en ]n∈h(x) and [en ]n∈h(y) are incomparable spaces. (2) if xE This will clearly finish the proof using the fact that subspaces of the same finite codimension in a common superspace are isomorphic.

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We will construct a partition of N into intervals I00 < I01 < I02 < I10 < I11 < I12 < · · · such that if we set Jn0 = In0 ∪ In2 and Jn1 = In1 , the following conditions hold: (1) for all n, |Jn0 | = |Jn1 |, sn−1 s (2) if s ∈ 2n , a = J0 0 ∪ J1s1 ∪ · · · ∪ Jn−1 ∪ In0 , and A ∈ [a, N], then for some i,   f (A)2i , f (A)2i+1 ⊆ In0 , s

s

n−1 ∪ In1 , and A ∈ [a, N], then for some i, (3) if s ∈ 2n , a = J0 0 ∪ J1s1 ∪ · · · ∪ Jn−1

  f (A)2i , f (A)2i+1 ⊆ In1 . Assuming this is done, for x ∈ 2N we set h(x) = J0 0 ∪ J1x1 ∪ · · · . Then for all n there is an i such that x

      f h(x) 2i , f h(x) 2i+1 ⊆ Jnxn . y

y

/ 0 y, then h(y) = J0 0 ∪ J1 1 ∪ · · · is disjoint from an infinite number of Jnxn and Therefore, if xE thus also from an infinite number of intervals [f (h(x))2i , f (h(x))2i+1 ], whence [en ]n∈h(x) does not embed into [en ]n∈h(y) . Similarly, [en ]n∈h(y) does not embed into [en ]n∈h(x) . On the other hand, if xE0 y, then clearly |h(x) \ h(y)| = |h(y) \ h(x)| < ∞. It therefore only remains to construct the intervals Ini . So suppose by induction that I00 < 0 1 2 < In−1 < In−1 have been chosen (the initial step being n = 0) such that I01 < I02 < · · · < In−1 0 2 + 1 = max In−1 + 1. For each s ∈ 2n and a = the conditions are satisfied. Let m = max Jn−1 sn−1 s0 s1 J0 ∪ J1 ∪ · · · ∪ Jn−1 , there are by continuity of f some ks > m, some interval m  Ms  ks and an integer is such that for all A ∈ [a ∪ [m, ks ], N], we have   f (A)2is , f (A)2is +1 = Ms . s

s

n−1 , we have for Let now k = maxs∈2n ks and In0 = [m, k]. Then if s ∈ 2n and a = J0 0 ∪ · · · ∪ Jn−1 0 all A ∈ [a ∪ In , N] some i such that

  f (A)2i , f (A)2i+1 ⊆ In0 . s

s

n−1 there are by continuity of f some Again for each s ∈ 2n and a = J0 0 ∪ J1s1 ∪ · · · ∪ Jn−1 ls > k + 1, some interval k + 1  Ls  ls and an integer js such that for all A ∈ [a ∪ [k + 1, ls ], N], we have

  f (A)2js , f (A)2js +1 = Ls .

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s

n−1 Let now l = maxs∈2n ls + k and In1 = [k + 1, l]. Then if s ∈ 2n and a = J0 0 ∪ · · · ∪ Jn−1 , we have 1 for all A ∈ [a ∪ In , N] some j such that

  f (A)2j , f (A)2j +1 ⊆ In1 . Finally, we simply let In2 = [l + 1, l + |In1 | − |In0 |]. This finishes the construction.

2

Definition 7.4. We define a quasi order ⊆∗ and a partial order ⊆0 on the space [N] of infinite subsets of N by the following conditions: A ⊆∗ B

⇔

A \ B is finite

and A ⊆0 B

⇔

  A = B or ∃n ∈ B \ A: A ⊆ B ∪ [0, n[ .

Also, if (an ) and (bn ) are infinite sequences of integers, we let (an ) ∗ (bn )

⇔

∀∞ n an  bn .

We notice that ⊆0 is a partial order refining the quasi order ⊆∗ , namely, whenever A ⊆∗ B we let A ⊆0 B if B ∗ A or A = B or A  B admits a greatest element which belongs to B. Proposition 7.5. (1) (2) (3) (4)

Any closed partial order on a Polish space Borel embeds into ⊆0 . Any partial order on a set of size at most ℵ1 embeds into ⊆0 . The quasi order ⊆∗ embeds into ⊆0 , but does not Borel embed. And finally ⊆0 Borel embeds into ⊆∗ .

Proof. (1) By an unpublished result of A. Louveau [29], any closed partial order on a Polish space Borel embeds into (P(N), ⊆). And if we let (Jn ) be a partition of N into countable many infinite subsets, we see that (P(N), ⊆) Borel embeds into ⊆∗ and ⊆0 by the mapping A →  n∈A Jn . (2) & (3) It is well known that any partial order of size at most ℵ1 embeds into ⊆∗ and if we let s : [N] → [N] be any function such that |A  B| < ∞ ⇔ s(A) = s(B) and |A  s(A)| < ∞, i.e., s is a selector for E0 , then s embeds ⊆∗ into ⊆0 . To see that there cannot be a Borel embedding of ⊆∗ into ⊆0 , we notice that if h : [N] → [N] was a Borel function such that A ⊆∗ B ⇔ h(A) ⊆0 h(B), then, in particular, |A  B| is finite ⇔ h(A) = h(B), contradicting that E0 is a non-smooth equivalence relation on [N]. (4) To see that ⊆0 Borel embeds into ⊆∗ , we define for an infinite subset A of N a sequence of integers g(A) = (an ) by an =

 i∈A∩[0,n]

2i .

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Suppose now that g(A) = (an ) and g(B) = (bn ). Then for each n, an = bn

⇔

A ∩ [0, n] = B ∩ [0, n]

and an < bn

⇔

∃m ∈ B \ A, m  n, A ∩ [0, n] ⊆ B ∪ [0, m[.

Thus, we have an = bn for infinitely many n if and only if A = B, and if an < bn for infinitely many n, then either B \ A is infinite or for some m ∈ B \ A we have A ⊆ B ∪ [0, m[. Moreover, if B \ A is infinite, then for infinitely many n, an < bn . So B ∗ A

(bn ) ∗ (an )

⇒

(B ∗ A or A ⊆0 B),

⇒

and thus by contraposition (bn ) ∗ (an )

⇒

B ⊆∗ A.

Also, if (bn ) ∗ (an ), then B ∗ A or A ⊆0 B, so if moreover B ⊆0 A, we would have A ⊆0 B and hence A = B, contradicting g(B) = (bn ) ∗ (an ) = g(A). Thus, B ⊆0 A

⇒

(bn ) ∗ (an ).

To see that also (bn ) ∗ (an )

⇒

B ⊆0 A,

notice that if (bn ) ∗ (an ) but B 0 A, then, as B ⊆∗ A, we must have A ⊆0 B and hence (an ) ∗ (bn ). But then an = bn for almost all n and thus A = B, contradicting B 0 A. Therefore, B ⊆0 A

⇔

(bn ) ∗ (an ),

and we thus have a Borel embedding of ⊆0 into the quasi order ∗ on the space NN . It is well known and easy to see that this latter Borel embeds into ⊆∗ and hence so does ⊆0 . 2 Proposition 7.6. Any Banach space without a minimal subspace contains a subspace with an F.D.D. (Fn ) satisfying one of the two following properties: (a) if A, B ⊆ N are infinite, then 

Fn 

n∈A



Fn

⇔

A ⊆∗ B,

Fn

⇔

A ⊆0 B.

n∈B

(b) if A, B ⊆ N are infinite, then  n∈A

Fn 

 n∈B

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Proof. Suppose X is a Banach space without a minimal subspace. Then by Theorem 3.13, we can find a continuously tight basic sequence (en ) in X. Using the infinite Ramsey theorem for analytic sets, we can also find an infinite set D ⊆ N such that (i) either for all infinite B ⊆ D, [ei ]i∈B embeds into its hyperplanes, (ii) or for all B ⊆ D, [ei ]i∈B is not isomorphic to a proper subspace. And, by Lemma 3.11, we can after renumbering the sequence (en )n∈D as (en )n∈N suppose that there is a continuous function f : [N] → [N] that for A, B ∈ [N], if B is disjoint from an infinite number of intervals [f (A)2i , f (A)2i+1 ], then [en ]n∈A does not embed into [en ]n∈B . We now construct a partition of N into intervals I0 < I1 < I2 < · · · such that the following conditions hold: – for all n, |I0 ∪ · · · ∪ In−1 | < |In |, – if A ∈ [N] and In ⊆ A, then for some i,   f (A)2i , f (A)2i+1 ⊆ In . Suppose by induction that I0 < I1 < · · · < In−1 have been chosen such that the conditions are satisfied. Let m = max In−1 + 1. For each a ⊆ [0, m[ there are by continuity of f some la > m, some interval m  Ma  la and an integer ia such that for all A ∈ [a ∪ [m, la ], N], we have   f (A)2ia , f (A)2ia +1 = Ma . Let now l > maxa⊆[0,m[ la be such that |I0 ∪ · · · ∪ In−1 | < l − m, and set In = [m, l[. Then if a ⊆ [0, m[, we have for all A ∈ [a ∪ In , N] some i such that   f (A)2i , f (A)2i+1 ⊆ In , which ends the construction.

n−1 ∗ i∈In . Clearly, i=0 dim Fi < dim Fn , and if A\B is infinite and we let A =  Let now Fn =∗ [ei ] ∗ then B will be disjoint from an infinite number of the intervals n∈A In and B = n∈B In , defined by f (A∗ ) and hence n∈A Fn = [en ]n∈A∗ does not embed into n∈B Fn = [en ]n∈B ∗ . In case of (i) we have that for all infinite C ⊆ N, (en )n∈C  (en )n∈C   (en )n∈C   (en )n∈C   · · · ,

where D  denotes D \ min D. So, in particular, for any infinite A ⊆ N, into all

n∈A Fn embeds of its finite codimensional subspaces and thus if A \ B is finite, then n∈A Fn  n∈B Fn . This gives us (a).

n−1

but B 0 A, we have, as dim Fn > In case (ii), if A ⊆0 B n∈A Fn embeds i=0 dim Fi , that F , then A \ B is finite and as a proper subspace of n∈B Fn . Conversely, if n∈A Fn  n∈B n

so either A ⊆0 B or B ⊆0 A. But if B ⊆0 A and A 0 B, then n∈B Fn embeds as a proper subspace into n∈A Fn and thus also into itself, contradicting (ii). Thus, A ⊆0 B. So assuming (ii) we have the equivalence in (b). 2

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We may observe that Tsirelson’s space satisfies case (a) of Proposition 7.6, while case (b) is verified by Gowers–Maurey’s space, or more generally by any space of type (1) to (4). By Propositions 7.5 and 7.6 we now have the following result. Theorem 7.7. Let X be an infinite-dimensional separable Banach space without a minimal subspace and let SB∞ (X) be the standard Borel space of infinite-dimensional subspaces of X ordered by the relation  of isomorphic embeddability. Then ⊆0 Borel embeds into SB∞ (X) and by consequence (a) any partial order of size at most ℵ1 embeds into SB∞ (X), (b) any closed partial order on a Polish space Borel embeds into SB∞ (X). We notice that this proves a strong dichotomy for the partial orders of Problem 7.1, namely, either they must be of size 1 or must contain any partial order of size at most ℵ1 and any closed partial order on a Polish space. In particular, in the second case we have well-ordered chains of length ω1 and also R-chains. This completes the picture of [13]. 8. Refining Gowers’ dichotomies We recall the list of inevitable classes of subspaces contained in a Banach space given by Gowers in [20]. Remember that a space is said to be quasi minimal if any two subspaces have a common -minorant, and strictly quasi minimal if it is quasi minimal but does not contain a minimal subspace. Also two spaces are incomparable in case neither of them embeds into the other, and totally incomparable if no space embeds into both of them. Theorem 8.1. (See Gowers [20].) Let X be an infinite-dimensional Banach space. Then X contains a subspace Y with one of the following properties, which are all possible and mutually exclusive. (i) Y is hereditarily indecomposable, (ii) Y has an unconditional basis such that any two disjointly supported block subspaces are incomparable, (iii) Y has an unconditional basis and is strictly quasi minimal, (iv) Y has an unconditional basis and is minimal. Here the condition of (ii) that any two disjointly supported block subspaces are incomparable, i.e., tightness by support, is equivalent to the condition that any two such subspaces are totally incomparable or just non-isomorphic. Theorem 1.1 improves the list of Gowers in case (iii). Indeed, any strictly quasi minimal space contains a tight subspace, but the space S(T (p) ), 1 < p < +∞ is strictly quasi minimal and not tight: it is saturated with subspaces of T (p) , which is strictly quasi minimal, and, as was already observed, it is not tight because its canonical basis is symmetric. Concerning case (i), properties of HI spaces imply that any such space contains a tight subspace, but it remains open whether every HI space with a basis is tight. Question 8.2. Is every HI space with a basis tight?

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Using Theorems 1.1 and 1.4, we refine the list of inevitable spaces of Gowers to 6 main classes as follows. Theorem 8.3. Let X be an infinite-dimensional Banach space. Then X contains a subspace Y with one of the following properties, which are all mutually exclusive. (1) Y is hereditarily indecomposable and has a basis such that any two block subspaces with disjoint ranges are incomparable, (2) Y is hereditarily indecomposable and has a basis which is tight and sequentially minimal, (3) Y has an unconditional basis such that any two disjointly supported block subspaces are incomparable, (4) Y has an unconditional basis such that any two block subspaces with disjoint ranges are incomparable, and is quasi minimal, (5) Y has an unconditional basis which is tight and sequentially minimal, (6) Y has an unconditional basis and is minimal. We conjecture that the space of Gowers and Maurey is of type (1), although we have no proof of this fact. Instead, in [14] we prove that an asymptotically unconditional HI space constructed by Gowers [18] is of type (1). We do not know whether type (2) spaces exist. If they do, they may be thought of as HI versions of type (5) spaces, i.e., of Tsirelson like spaces, so one might look for an example in the family initiated by the HI asymptotically 1 space of Argyros and Deliyanni, whose “ground” space is a mixed Tsirelson’s space based on the sequence of Schreier families [2]. The first example of type (3) was built by Gowers [17] and further analysed in [22]. Other examples are constructed in [14]. Type (4) means that for any two block subspaces Y and Z with disjoint ranges, Y does not embed into Z, but some further block subspace Y  of Y does (Y  therefore has disjoint support but not disjoint range from Z). It is unknown whether there exist spaces of type (4). Gowers sketched the proof of a weaker result, namely the existence of a strictly quasi minimal space with an unconditional basis and with the Casazza property, i.e., such that for no block sequence the sequence of odd vectors is equivalent to the sequence of even vectors, but his example was never actually checked. Alternatively, results of [26, Section 4] suggest that a mixed Tsirelson space example might be looked for. The main example of a space of type (5) is Tsirelson’s space. Actually since spaces of type (1) to (4) are either HI or satisfy the Casazza property, they are never isomorphic to a proper subspace. Therefore, for example, spaces with a basis saturated with block subspaces isomorphic to their hyperplanes must contain a subspace of type (5) or (6). So our results may reinforce the idea that Tsirelson’s space is the canonical example of classical space without a minimal subspace. It is worth noting that as a consequence of the theorem of James, spaces of type (3), (4) and (5) are always reflexive. Using some of the additional dichotomies, one can of course refine this picture even further. We shall briefly consider how this can be done using the 5th dichotomy plus a stabilisation theorem of A. Tcaciuc [41] generalising a result of [15]. We state a slightly stronger version of the theorem of Tcaciuc than what is proved in his paper and also point out that there is an unjustified use of a recent result of Junge, Kutzarova and Odell in his paper; their result only holds for 1  p < ∞. Tcaciuc’s theorem states that any Banach

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space contains either a strongly asymptotically p subspace, 1  p  +∞, or a subspace Y such that ∀M ∃n ∀U1 , . . . , U2n ⊆ Y ∃xi ∈ SUi (x2i−1 )ni=1 M (x2i )ni=1 , where the Ui range over infinite-dimensional subspaces of Y . The second property in this dichotomy will be called uniform inhomogeneity. As strongly asymptotically p bases are unconditional, while the HI property is equivalent to uniform inhomogeneity with n = 2 for all M, Tcaciuc’s dichotomy is only relevant for spaces with an unconditional basis. When combining Theorem 8.3, Tcaciuc’s result, Proposition 4.2 (see also [9]), the 5th dichotomy, the fact that asymptotically ∞ spaces are locally minimal, and the classical theorem of James, we obtain 19 inevitable classes of spaces and examples for 8 of them. The class (2) is divided into two subclasses and the class (4) into four subclasses, which are not made explicit here for lack of an example of space of type (2) or (4) to begin with. Recall that the spaces contained in any of the 12 subclasses of type (1)–(4) are never isomorphic to their proper subspaces, and in this sense these subclasses may be labeled “exotic”. On the contrary “classical”, “pure” spaces must belong to one of the 7 subclasses of type (5)–(6). Theorem 8.4. Any infinite-dimensional Banach space contains a subspace with a basis of one of the following types: Type

Properties

Examples

(1a) (1b)

HI, tight by range and with constants HI, tight by range, locally minimal

? G∗

(2)

HI, tight, sequentially minimal

?

(3a) (3b) (3c) (3d)

tight by support and with constants, uniformly inhomogeneous tight by support, locally minimal, uniformly inhomogeneous tight by support, strongly asymptotically p , 1  p < ∞ tight by support, strongly asymptotically ∞

? G∗u Xu Xu∗

(4)

unconditional basis, quasi minimal, tight by range

?

(5a)

?

(5d)

unconditional basis, tight with constants, sequentially minimal, uniformly inhomogeneous unconditional basis, tight, sequentially and locally minimal, uniformly inhomogeneous tight with constants, sequentially minimal, strongly asymptotically p , 1  p < ∞ tight, sequentially minimal, strongly asymptotically ∞

?

(6a) (6b) (6c)

unconditional basis, minimal, uniformly inhomogeneous minimal, reflexive, strongly asymptotically ∞ isomorphic to c0 or lp , 1  p < ∞

S T∗ c0 , p

(5b) (5c)

? T , T (p)

We know of no space close to being of type (5a) or (5b). A candidate for (5d) could be the dual of some partly modified mixed Tsirelson’s space not satisfying the blocking principle

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(see [26]). Schlumprecht’s space S [39] does not contain an asymptotically p subspace, therefore it contains a uniformly inhomogeneous subspace, which implies by minimality that S itself is of type (6a). The definition and analysis of the spaces G∗ , G∗u , Xu and Xu∗ can be found in [14]. For completeness we should mention that R. Wagner has also proved a dichotomy between asymptotic unconditionality and a strong form of the HI property [43]. His result could be used to further refine the cases of type (1) and (2). 9. Open problems Problem 9.1. (1) (2) (3) (4)

(5) (6) (7) (8) (9) (10) (11)

Does there exist a tight Banach space admitting a basis which is not tight? Does there exist a tight, locally block minimal and unconditional basis? Find a locally minimal and tight Banach space with finite cotype. Does there exist a tight Banach space which does not contain a basic sequence that is either tight by range or tight with constants? In other words, does there exist a locally and sequentially minimal space without a minimal subspace? Suppose [en ] is sequentially minimal. Does there exist a block basis all of whose subsequences are subsequentially minimal? Is every HI space with a basis tight? Is every tight basis continuously tight? Do there exist spaces of type (2), (4), (5a), (5b), (5d)? Suppose (en ) is tight with constants. Does (en ) have a block sequence that is (strongly) asymptotically p for some 1  p < ∞? Does there exist a separable HI space X such that ⊆∗ Borel embeds into SB∞ (X)? If X is a separable Banach space without a minimal subspace, does ⊆∗ Borel embed into SB∞ (X)? What about more complicated quasi orders, in particular, the complete analytic quasi order Σ 1 ? 1

References [1] G. Androulakis, N. Kalton, A. Tcaciuc, On Banach spaces containing p or c0 , preprint. [2] S. Argyros, I. Deliyanni, Examples of asymptotic 1 Banach spaces, Trans. Amer. Math. Soc. 349 (1997) 973–995. [3] S. Argyros, I. Deliyanni, D. Kutzarova, A. Manoussakis, Modified mixed Tsirelson spaces, J. Funct. Anal. 159 (1998) 43–109. [4] J. Bagaria, J. López-Abad, Weakly Ramsey sets in Banach spaces, Adv. Math. 160 (2) (2001) 133–174. [5] A. Brunel, L. Sucheston, On B-convex Banach spaces, Math. Systems Theory 7 (4) (1974) 294–299. [6] P.G. Casazza, W.B. Johnson, L. Tzafriri, On Tsirelson’s space, Israel J. Math. 47 (2–3) (1984) 81–98. [7] P.G. Casazza, N.J. Kalton, Unconditional bases and unconditional finite-dimensional decompositions in Banach spaces, Israel J. Math. 95 (1996) 349–373. [8] P.G. Casazza, T. Shura, Tsirelson’s Space, Lecture Notes in Math., vol. 1363, Springer-Verlag, Berlin, 1989. [9] S. Dilworth, V. Ferenczi, D. Kutzarova, E. Odell, On strongly asymptotically p spaces and minimality, J. London Math. Soc. (2) 75 (2007) 409–419. [10] V. Ferenczi, Minimal subspaces and isomorphically homogeneous sequences in a Banach space, Israel J. Math. 156 (2006) 125–140. [11] V. Ferenczi, A. Louveau, C. Rosendal, The complexity of classifying separable Banach spaces up to isomorphism, J. London Math. Soc., in press. [12] V. Ferenczi, C. Rosendal, Ergodic Banach spaces, Adv. Math. 195 (1) (2005) 259–282. [13] V. Ferenczi, C. Rosendal, Complexity and homogeneity in Banach spaces, in: Beata Randrianantoanina, Narcisse Randrianantoanina (Eds.), Banach Spaces and Their Applications in Mathematics, Walter de Gruyter, Berlin, 2007, pp. 83–110.

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[14] V. Ferenczi, C. Rosendal, Banach spaces without minimal subspaces – Examples, preprint. [15] T. Figiel, R. Frankiewicz, R.A. Komorowski, C. Ryll-Nardzewski, Selecting basic sequences in φ-stable Banach spaces, Dedicated to Professor Aleksander Pełczy´nski on the occasion of his 70th birthday, Studia Math. 159 (3) (2003) 499–515. [16] T. Figiel, J. Lindenstrauss, V.D. Milman, The dimension of almost spherical sections of convex bodies, Acta Math. 139 (1977) 53–94. [17] W.T. Gowers, A solution to Banach’s hyperplane problem, Bull. London Math. Soc. 26 (6) (1994) 523–530. [18] W.T. Gowers, A hereditarily indecomposable space with an asymptotic unconditional basis, in: Geometric Aspects of Functional Analysis, Israel, 1992–1994, in: Oper. Theory Adv. Appl., vol. 77, Birkhäuser, Basel, 1995, pp. 112– 120. [19] W.T. Gowers, A new dichotomy for Banach spaces, Geom. Funct. Anal. 6 (6) (1996) 1083–1093. [20] W.T. Gowers, An infinite Ramsey theorem and some Banach space dichotomies, Ann. of Math. (2) 156 (3) (2002) 797–833. [21] W.T. Gowers, B. Maurey, The unconditional basic sequence problem, J. Amer. Math. Soc. 6 (4) (1993) 851–874. [22] W.T. Gowers, B. Maurey, Banach spaces with small spaces of operators, Math. Ann. 307 (1997) 543–568. [23] W.B. Johnson, A reflexive Banach space which is not sufficiently Euclidean, Studia Math. 55 (1978) 201–205. [24] A.S. Kechris, Classical Descriptive Set Theory, Springer-Verlag, New York, 1995. [25] J.L. Krivine, Sous-espaces de dimension finie des espaces de Banach réticulés, Ann. of Math. (2) 104 (1) (1976) 1–29. [26] D. Kutzarova, D.H. Leung, A. Manoussakis, W.K. Tang, Minimality properties of Tsirelson type spaces, Studia Math. 187 (3) (2008) 233–263. [27] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces, Springer-Verlag, New York, Heidelberg, Berlin, 1979. [28] J. López-Abad, Coding into Ramsey sets, Math. Ann. 332 (4) (2005) 775–794. [29] A. Louveau, Closed orders and their vicinity, preprint, 2001. [30] B. Maurey, V. Milman, N. Tomczak-Jaegermann, Asymptotic infinite-dimensional theory of Banach spaces, in: Geometric Aspects of Functional Analysis, Israel, 1992–1994, in: Oper. Theory Adv. Appl., vol. 77, Birkhäuser, Basel, 1995, pp. 149–175. [31] E. Odell, T. Schlumprecht, The distortion problem, Acta Math. 173 (2) (1994) 259–281. [32] E. Odell, T. Schlumprecht, On the richness of the set of p’s in Krivine’s theorem, in: Geometric Aspects of Functional Analysis, Israel, 1992–1994, in: Oper. Theory Adv. Appl., vol. 77, Birkhäuser, Basel, 1995, pp. 177–198. [33] E. Odell, T. Schlumprecht, A Banach space block finitely universal for monotone bases, Trans. Amer. Math. Soc. 352 (4) (2000) 1859–1888. [34] E. Odell, T. Schlumprecht, Trees and branches in Banach spaces, Trans. Amer. Math. Soc. 354 (10) (2002) 4085– 4108. [35] A.M. Pełczar, Subsymmetric sequences and minimal spaces, Proc. Amer. Math. Soc. 131 (3) (2003) 765–771. [36] A. Pełczy´nski, Universal bases, Studia Math. 32 (1969) 247–268. [37] C. Rosendal, Incomparable, non-isomorphic and minimal Banach spaces, Fund. Math. 183 (3) (2004) 253–274. [38] C. Rosendal, Infinite asymptotic games, Ann. Inst. Fourier, in press. [39] T. Schlumprecht, An arbitrarily distortable Banach space, Israel J. Math. 76 (1–2) (1991) 81–95. [40] T. Schlumprecht, unpublished notes. [41] A. Tcaciuc, On the existence of asymptotic-p structures in Banach spaces, Canad. Math. Bull. 50 (4) (2007) 619– 631. [42] B.S. Tsirelson, Not every Banach space contains p or c0 , Funct. Anal. Appl. 8 (1974) 138–141. [43] R. Wagner, Finite high-order games and an inductive approach towards Gowers’s dichotomy, in: Proceedings of the International Conference “Analyse & Logique”, Mons, 1997, Ann. Pure Appl. Logic 111 (1–2) (2001) 39–60.

Journal of Functional Analysis 257 (2009) 194–218 www.elsevier.com/locate/jfa

Orbits in symmetric spaces ✩ F. Sukochev a,∗ , D. Zanin b a School of Mathematics and Statistics, University of New South Wales, Kensington, NSW 2052, Australia b School of Computer Science, Engineering and Mathematics, Flinders University, Bedford Park, SA 5042, Australia

Received 10 November 2008; accepted 21 January 2009 Available online 20 February 2009 Communicated by N. Kalton

Abstract We characterize those elements in fully symmetric spaces on the interval (0, 1) or on the semi-axis (0, ∞) whose orbits are the norm-closed convex hull of their extreme points. © 2009 Elsevier Inc. All rights reserved. Keywords: Symmetric function spaces; Orbits; Closed convex hull of extreme points

1. Introduction The following semigroups of bounded linear operators play a fundamental role in the interpolation theory of linear operators for the couple (L1 , L∞ ) of Lebesgue measurable functions on intervals (0, 1) and (0, ∞). The semigroup of absolute contractions, or admissible operators (see e.g. [10, II.3.4])     Σ := T : L1 + L∞ → L1 + L∞ : max T L1 →L1 , T L∞ →L∞  1 , the semigroup of substochastic operators (see e.g. [2, p. 107]) Σ+ := {0  T ∈ Σ} ✩

Research was partially supported by the ARC.

* Corresponding author.

E-mail addresses: [email protected] (F. Sukochev), [email protected] (D. Zanin). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.01.025

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195

and, in the case of the interval (0, 1), the semigroup of doubly stochastic operators  Σ  := 0  T ∈ Σ+ :

1



1 (T x)(s) ds =

0

x(s) ds, ∀x  0, T 1 = 1 0

(see e.g. [13]). If x ∈ L1 + L∞ (respectively, 0  x ∈ L1 + L∞ or 0  x ∈ L1 (0, 1)) we denote by Ω(x) (respectively Ω+ (x) and Ω  (x)) the orbit of x with respect to the semigroups Σ (respectively, Σ+ , and Σ  ). A Banach function space E (on (0, 1) or (0, ∞), see [2, pp. 2–3]) is called an exact interpolation space if every T ∈ Σ maps E into itself and T E→E  1, or alternatively, if Ω(x) ⊂ E and yE  xE for every x ∈ E and every y ∈ Ω(x). The class of exact interpolation spaces admits an equivalent description in terms of (sub)majorization in the sense of Hardy, Littlewood and Polya. Recall, that if x, y ∈ L1 + L∞ , then y is said to be a submajorized by x in the sense of Hardy, Littlewood and Polya, written y ≺≺ x if and only if t

∗

t

y (s) ds  0

x ∗ (s) ds,

t  0.

0

Here, x ∗ denotes the non-increasing right-continuous rearrangement of x given by     x ∗ (t) = inf s  0: m |x|  s  t and m is Lebesgue measure. If 0  x, y ∈ L1 , then we say that y is majorized by x (written y ≺ x) if and only if y ≺≺ x and y1 = x1 . A Banach function space E is said to be fully symmetric if and only if x ∈ E, y ∈ L1 + L∞ y ≺≺ x ⇒ y ∈ E and yE  xE . The classical Calderon– Mityagin theorem (see [4,10,2]) gives an alternative description of the sets Ω(x), x ∈ L1 + L∞ and Ω+ (x), 0  x ∈ L1 + L∞ as follows Ω(x) = {y ∈ L1 + L∞ : y ≺≺ x},

Ω+ (x) = {0  y ∈ L1 + L∞ : y ≺≺ x}

and (in the case of the interval (0, 1) and 0  x ∈ L1 (0, 1)) Ω  (x) = {0  y ∈ L1 : y ≺ x}, which shows, in particular, that the classes of exact interpolation spaces and fully symmetric spaces coincide. Let fully symmetric Banach function space E be fixed. The principal aim of the paper is to give conditions for a given 0  x ∈ E which are necessary and sufficient for each of the sets Ω+ (x), Ω  (x) to be the norm closure of the convex hull of their extreme points. If E = L1 (0, 1), then it has been shown by Ryff (see [13]) that if 0  x ∈ E, then the orbit Ω  (x) is weakly compact and so, due to the Krein–Milman theorem, the orbit Ω  (x) is the weak (and hence norm)-closed convex hull of its extreme points. It follows from the results of [7] that the set Ω  (x) is weakly compact in any separable symmetric space E. Hence, Ω  (x) is the weak (and hence norm)-closed convex hull of its extreme points in any separable symmetric space E. If a fully symmetric space E is not separable, then it is not the case in general that orbits are weakly compact. A trivial example yields the orbit Ω(χ[0,1] ) in fully symmetric non-separable

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space L∞ (0, 1). Indeed, it is obvious that this orbit coincides with the unit ball of L∞ (0, 1) and the latter is not weakly compact since the space L∞ (0, 1) is non-reflexive. Nonetheless, it is an interesting question to give necessary and sufficient conditions that the orbit of a given element should be the norm-closed convex hull of its extreme points. This question was considered by Braverman and Mekler (see [3]) in the case of the interval (0, 1) and orbits Ω(x). They showed that for every fully symmetric space E on (0, 1) satisfying the condition lim

τ →∞

1 στ E→E = 0 τ

(1)

that Ω(x) is indeed the norm-closed convex hull of the set of its extreme points, for every x ∈ E (see [3, Theorem 3.1]). Here στ denotes the usual dilation operator (see the following section for definition and properties). They showed as well that the converse assertion is valid in case that E is a Marcinkiewicz space on (0, 1). As explained below, this converse assertion, however, fails for arbitrary fully symmetric spaces. We show (Theorem 21) that if E is a fully symmetric space on (0, 1) and if 0  x ∈ E, then Ω  (x) is the norm-closed convex hull of its extreme points if and only if ϕ(x) := lim

τ →∞

1 στ (x ∗ ) = 0. E τ

(2)

As shown in Corollary 27 this implies the result of Braverman and Mekler. In Appendix A, we demonstrate that the conditions (1) and (2) are distinct in the class of Orlicz spaces. If E is an Orlicz space, then it is the case that (2) holds, and so by Theorems 21 and 22 for every 0  x ∈ E, the sets Ω  (x), Ω+ (x) and Ω(x) are the norm-closed convex hulls of its extreme points. However, there are non-separable Orlicz spaces E which fail condition (1). In Appendix A, we also introduce the notion of symmetric and fully symmetric functionals. The latter are a “commutative” counterpart of Dixmier traces appeared in non-commutative geometry (see e.g. [5]). Symmetric and fully symmetric functionals are extensively studied in [8,9] (see also [5] and references therein). Note, however, that our terminology differs from that used in just cited articles. A subclass of Marcinkiewicz spaces admitting symmetric functionals which fail to be fully symmetric is described in [9]. It follows from our results that any symmetric functional on a fully symmetric space satisfying (2) is automatically fully symmetric. In particular, this implies that an Orlicz space does not possess any singular symmetric functionals (see Proposition 34). This latter result strengthens the result of [8, Theorem 3.1] that an Orlicz space does not possess any singular fully symmetric functionals. Results similar to Theorems 21 and 22 hold also for fully symmetric spaces E on the semi-axis (see Theorems 23–26). The main results of this article are contained in Section 4. In the following section we present some definitions from the theory of symmetric spaces, as some of our results hold in a slightly more general setting than that of fully symmetric spaces. For more details on the latter theory we refer to [10,11,2]. Section 3 treats various properties of the functional ϕ and its modifications needed in Section 4. We would like to emphasize the difference between geometric properties of the orbit Ω(x) and those of Ω  (x) and Ω+ (x). This is especially noticeable in the description of the respective sets of their extreme points. The extreme points of the sets Ω(x) and Ω  (x), x  0 are well known (see [14,6]) and are given by   extr Ω(x) = {y ∈ L1 + L∞ : y ∗ = x ∗ },

  extr Ω  (x) = {0  y ∈ L1 : y ∗ = x ∗ },

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whereas the description of extreme points of the set Ω+ (x), x  0 given by   extr Ω+ (x) = {0  y ∈ L1 + L∞ : y ∗ = x ∗ χ[0,β] for some β  ∞} when x ∗ (∞) := limt→∞ x ∗ (t) = 0, and by   extr Ω+ (x) = {0  y ∈ L1 + L∞ : y ∗ = x ∗ χ[0,β] for some β  ∞ and yχ{y 0, is somewhat less known, so we present in Appendix A a careful exposition of the latter equalities. 2. Preliminaries Let L0 be a space of Lebesgue measurable functions either on (0, 1) or (0, ∞) finite almost everywhere (with identification m-a.e.). Here m is a Lebesgue measure. Define S0 as the subset of L0 which consists of all functions x such that m({|x| > s}) is finite for some s. Let E be a Banach space of real-valued Lebesgue measurable functions either on (0, 1) or (0, ∞) (with identification m-a.e.). E is said to be ideal lattice if x ∈ E and |y|  |x| implies that y ∈ E and yE  xE . The ideal lattice E ⊆ S0 is said to be symmetric space if for every x ∈ E and every y the assumption y ∗ = x ∗ implies that y ∈ E and yE = xE . If E = E(0, 1) is a symmetric space on (0, 1), then L∞ ⊆ E ⊆ L1 . If E = E(0, ∞) is a symmetric space on (0, ∞), then L1 ∩ L ∞ ⊆ E ⊆ L 1 + L ∞ . Symmetric space E is said to be fully symmetric if and only if x ∈ E, y ∈ L1 + L∞ y ≺≺ x ⇒ y ∈ E and yE  xE . We now gather some additional terminology from the theory of symmetric spaces that will be needed in the sequel. Suppose E is a symmetric space. Following [3], E will be called strictly symmetric if and only if whenever x, y ∈ E and y ≺≺ x then yE  xE . It is clear that if E is fully symmetric then E is strictly symmetric, but the converse assertion is not valid. The norm  · E is called Fatou norm if, for every sequence xn ↑ x ∈ E, it follows that xn E ↑ xE . This is equivalent to the assertion that the unit ball of E is closed with respect to almost everywhere convergence. It is well known that if the norm on E is a Fatou norm then E is strictly symmetric. If τ > 0, the dilation operator στ is defined by setting (στ (x))(s) = x( τs ), s > 0 in the case of the semi-axis. In the case of the interval (0, 1) the operator στ is defined by

(στ x)(s) =

x(s/τ ), 0,

s  min{1, τ }, τ < s  1.

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The operators στ (τ  1) satisfy semigroup property στ1 στ2 = στ1 τ2 . If E is a symmetric space and if τ > 0, then the dilation operator στ is a bounded operator on E and στ E→E  max{1, τ }. We will need also the notion of a partial averaging operator (see [3]). and denote Let A = {Ak } be a (finite or infinite) sequence of disjoint sets of finite measure  by A the collection of all such sequences. Denote by A∞ the complement of k Ak . The partial averaging operator is defined by  1  P (x|A) = x(s) ds χAk + xχA∞ . m(Ak ) k

Ak

Note, that we do not require A∞ to have a finite measure. Every partial averaging operator is a contraction both in L1 and L∞ . Hence, P (·|A) is also a contraction in E. In case of the interval (0, 1), P (·|A) is a doubly stochastic operator in the sense of [13]. Since P (·|A) ∈ Σ , then P (x|A) ∈ Ω(x) (respectively, P (x|A) ∈ Ω  (x) if x ∈ L1 ) for every A ∈ A. As will be seen, elements of the form P (x|A) play a central role. The following properties of rearrangements can be found in [10]. If x, y ∈ L1 + L∞ , then (x + y)∗ ≺≺ x ∗ + y ∗

(3)

(x ∗ − y ∗ ) ≺≺ (x − y)∗ .

(4)

and

Let us recall some classical examples of fully symmetric spaces. Let ψ be a concave increasing continuous function. The Marcinkiewicz space Mψ is the linear space of those functions x ∈ S0 , for which xMψ

1 = sup t ψ(t)

t

x ∗ (s) ds < ∞.

0

Equipped with the norm xMψ , Mψ is a fully symmetric space with Fatou norm. Let M(t) be a convex function on [0, ∞) such that M(t) > 0 for all t > 0 and such that 0 = M(0) = lim

t→0

M(t) t = lim . t→∞ M(t) t

Denote by LM the Orlicz space on [0, ∞) (see e.g. [11,10]) endowed with the norm 

∞

xLM = inf λ: λ > 0,

      M x(t) /λ dt  1 .

0

Equipped with the norm xLM , LM is a fully symmetric space with Fatou norm.

(5)

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For further properties of Marcinkiewicz and Orlicz spaces, we refer to [10–12]. For 0  x ∈ L1 + L∞ , we set    Q+ (x) = Conv extr Ω+ (x) . For 0  x ∈ L1 , we set    Q (x) = Conv extr Ω  (x) . For 0  x ∈ L1 + L∞ , we set Q (x) = Conv{y ∗ = x ∗ , yχ{y
(6)

2

Proof. Fix ε > 0. It follows from [10, II.2.1], t



∗

x (s) ds  ε +

t x(s) ds,

e1

0

y ∗ (s) ds  ε +

 y(s) ds e2

0

for some e1 and e2 with m(ei ) = t. However, 

 x(s) ds + e1

 y(s) ds 

e2

(x + y)(s) ds

e1 ∪e2

  sup m(e)=2t

Note, that

 2t 0

u(s) ds =

t

0 (2σ 12 u)(s) ds.

2t (x + y)(s) ds =

e

2

Lemma 2. If x, y ∈ L1 + L∞ and y ≺≺ x, then, ∗  στ (y)  στ (y ∗ ) ≺≺ στ (x ∗ ).

0

(x + y)∗ (s) ds.

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Proof. Set dy (s) = m(t: |y(t)| > s). In the case of the semi-axis, dστ y = τ dy = dστ (y ∗ ) . In the case of the interval (0, 1), dστ y  τ dy and dστ (y ∗ ) = min{1, τ dy }. Hence, dστ y  dστ (y ∗ ) and so (στ (y))∗  στ (y ∗ ). Finally, t

t

τ

∗

στ (y )(s) ds = τ 0

t

τ

∗

y (s) ds  τ 0

t

∗

x (s) ds = 0

στ (x ∗ )(s) ds.

2

0

The next lemma introduces the dilation functional ϕ on E, which is a priori non-linear. The behavior of the functional ϕ on the positive part E+ of E provides the key to our main question. Lemma 3. For every x ∈ E the following limit exists and is finite. 1 σs (x ∗ ) , E s→∞ s

ϕ(x) = lim

x ∈ E.

(7)

If, in addition, E = E(0, ∞), then the following limits exist and are finite. 1 σs (x ∗ )χ[0,1] , E s 1 ϕcut (x) = lim σs (x ∗ )χ[0,s] E , s→∞ s ϕfin (x) = lim

s→∞

x ∈ E,

(8)

x ∈ E.

(9)

The following properties hold. (i) (ii) (iii) (iv) (v) (vi)

If E If E If E If E If E If E

is symmetric, then ϕ(y)  ϕ(x) provided that x, y ∈ E satisfy y ∗  x ∗ . is symmetric, then ϕ(x)  xE for every x ∈ E. is strictly symmetric, then ϕ(y)  ϕ(x) provided that x, y ∈ E satisfy y ≺≺ x. is symmetric, then ϕ(στ (x ∗ )) = τ ϕ(x), τ > 0. is strictly symmetric, then ϕ is norm-continuous. is strictly symmetric, then ϕ is convex.

If, in addition, E = E(0, ∞), then ϕfin also satisfies (i)–(vi), while ϕcut satisfies (i), (ii), (iii), (v) and (vi). If, in addition, E  L1 , then ϕcut also satisfies (iv). Proof. We prove that the function s → 1s σs x ∗ E is decreasing. Let s2 > s1 . We have s2 = s3 s1 and s3 > 1. Therefore,   1 σs σs (x ∗ )  σs3 E→E σs (x ∗ )  1 σs (x ∗ ) , 3 1 1 1 E E E s2 s2 s1 since σs3 E→E  s3 . It follows immediately that the limit in (7) exists. (i) Trivial. (ii) This follows from the fact that σs (x ∗ )E  sxE . (iii) Since y ≺≺ x, it follows that σs (y ∗ ) ≺≺ σs (x ∗ ). Since E is strictly symmetric, it follows that σs (y ∗ )E  σs (x ∗ )E . Therefore, 1 σs (y ∗ )  lim 1 σs (x ∗ ) = ϕ(x). E E s→∞ s s→∞ s

ϕ(y) = lim

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201

(iv) Applying the semigroup property of the dilation operators στ ,   1 σs στ (x ∗ ) = τ lim 1 σsτ (x ∗ ) = τ ϕ(x). E E s→∞ s τ →∞ sτ lim

(v) By triangle inequality,    σs (x ∗ ) − σs (y ∗ )   σs (x ∗ − y ∗ ) . E E E Using (4) and Lemma 2 one can obtain σs (x ∗ − y ∗ ) ≺≺ σs ((x − y)∗ ). Since E is strictly symmetric,      σs (x ∗ ) − σs (y ∗ )   σs (x − y)∗ . E

E

E

Now, one can divide by s and let s → ∞. Therefore,   ϕ(x) − ϕ(y)  ϕ(x − y)  x − yE . (vi) It follows from (3) and Lemma 2 that σs ((x + y)∗ ) ≺≺ σs (x ∗ ) + σs (y ∗ ). Therefore, ϕ(x + y) = lim

s→∞

    1 σs (x + y)∗  lim 1 σs (x ∗ ) + σs (y ∗ ) = ϕ(x) + ϕ(y). E E E s→∞ s s

Existence and properties (i)–(vi) of ϕfin can be proved in a similar way. Existence and properties (i), (ii), (iii), (iv), (vi) of ϕcut can be proved in a similar way. Let us prove (iv) for ϕcut . (iv) Assume E ⊂ L1 . By Lemma 4 below, ϕ(x ∗ χ[τ −1 ,1] ) = ϕcut (x ∗ χ[τ −1 ,1] ) = 0. Hence, ϕ(x ∗ χ[0,τ −1 ] )  ϕ(x ∗ χ[0,1] )  ϕ(x ∗ χ[0,τ −1 ] ) + ϕ(x ∗ χ[τ −1 ,1] ) = ϕ(x ∗ χ[0,τ −1 ] ). Therefore,     ϕcut στ (x ∗ ) = ϕ στ (x ∗ χ[0,τ −1 ] ) = τ ϕ(x ∗ χ[0,τ −1 ] ) = τ ϕcut (x).

2

Lemma 4. If E = E(0, 1) be a symmetric space on (0, 1) and x ∈ L∞ , then ϕ(x) = 0. If E = E(0, ∞) be a symmetric space on (0, ∞) and x ∈ L∞ ∩ E, then ϕfin (x) = 0. If E = E(0, ∞)  L1 and x ∈ E ∩ L∞ , then ϕcut (x) = 0. In particular, the functional ϕ vanishes on every separable space E = E(0, 1). Proof. Clearly, ϕ(x) = ϕ(x ∗ χ[0,1] )  x∞ ϕ(χ[0,1] ) in the first case. Similarly, ϕfin (x)  x∞ ϕfin (χ[0,1] ) (ϕcut (x)  x∞ ϕcut (χ[0,1] )) in the second (third) case. It is clear that ϕ(χ[0,1] ) = 0 (ϕfin (χ[0,1] ) = 0) in the first (second) case. Also, E ⊂ L1 implies that χ[0,n] E = o(n) and, therefore, ϕcut (χ[0,1] ) = 0. The assertion follows immediately. 2 Lemma 5. Let E be a strictly symmetric space. For functions 0  x1 , . . . , xk ∈ E and numbers λ 1 , . . . , λk  0  ϕ

k i=1

 λi xi = ϕ



k i=1

 λi xi∗

.

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If E = E(0, ∞), then the same is valid for ϕfin . If, in addition, E  L1 , then the same is valid for ϕcut . Proof. Applying the inequality (6) n times, we have for positive functions x1 , . . . , x2n  ∗  x1 + · · · + x2∗n ≺≺ 2n σ2−n (x1 + · · · + x2n ). Therefore, by Lemma 3(iii),     ϕ x1∗ + · · · + x2∗n  ϕ 2n σ2−n (x1 + · · · + x2n ) . By Lemma 3(iv), 2k ϕ(σ2−k (z∗ )) = ϕ(z∗ ). Therefore,   ϕ x1∗ + · · · + x2∗n  ϕ(x1 + · · · + x2n ). Converse inequality follows trivially from (3) and Lemma 3(iii). The assertion of the lemma follows now from Lemma 3(v). 2 Note, that y and z in the proposition below are arbitrary, that is, y, z do not necessary belong to Q+ (x). Proposition 6. Let E be a symmetric space equipped with a Fatou norm. If x  0 ∈ E, then in each of the following cases there exists a decomposition x = y + z, such that y, z  0 and such that the following assertions hold. (i) (ii) (iii) (iv)

If E = E(0, 1), then ϕ(x) = ϕ(y) = ϕ(z). If E = E(0, ∞) and ϕcut (x) = 0, then ϕ(x) = ϕ(y) = ϕ(z). If E = E(0, ∞), then ϕfin (x) = ϕfin (y) = ϕfin (z). If E = E(0, ∞), then ϕcut (x) = ϕcut (y) = ϕcut (z).

Proof. We will prove only the first assertion. The proofs of the third and fourth assertions are exactly the same. The proof of the second assertion requires replacement of the interval [ m1 , n1 ] with the interval [n, m]. We may assume that x = x ∗ . Fix n ∈ N . The sequence σn (xχ[ 1 , 1 ] ) converges to σn (xχ[0, 1 ] ) m n n almost everywhere when m → ∞. By the definition of Fatou norm, σn (xχ

→m σn (xχ 1 ) . [0, ] E

1 1 ) [m ,n] E

n

For each n ∈ N , one can select f (n) > n, such that 

1 σn (xχ 1 ) . 1 [ f (n) , n1 ] ) E  1 − n [0, n ] E

σn (xχ

Fix some n0 and set nk = f k (n0 ), k ∈ N . Here, f k = f ◦ · · · ◦ f (k times). Define

F. Sukochev, D. Zanin / Journal of Functional Analysis 257 (2009) 194–218

y=

∞

xχ[

k=0

z=

∞

xχ[

k=1

1 1 n2k+1 , n2k

203

],

1 1 n2k , n2k−1 ]

.

It is clear, that 1 σn (y ∗ )  1 σn (y)  1 σn (xχ 1 1 ) . 2k 2k 2k [ , ] E E E n2k+1 n2k n2k n2k n2k

(10)

By definition of nk ,

 1 1 1 σn (xχ 1 σn (xχ 1 ) .  1− 1 ) 2k 2k [ , ] [0, ] E E n n n n2k n2k n2k 2k+1 2k 2k

(11)

It follows from (10) and (11) that  

1 1 σn (y ∗ )  1 − 1 σn (xχ 1 )  1 − 1 ϕ(xχ 1 ). 2k 2k [0, n ] E [0, n ] E n2k n2k n2k n2k 2k 2k By Lemma 4, ϕ(xχ[

1 n2k

ϕ(xχ[0,

,1] ) = 0.

1 n2k

(12)

Since ϕ is convex, then

] )  ϕ(x)  ϕ(xχ[0, n1 ] ) + ϕ(xχ[ n1 ,1] ) = ϕ(xχ[0, n1 ] ). 2k

2k

(13)

2k

It follows from (12) and (13) that 

1 σn (y ∗ )  1 − 1 ϕ(x). 2k E n2k n2k Passing to the limit, we obtain ϕ(y)  ϕ(x). The converse inequality is obvious. Hence, ϕ(y) = ϕ(x) = ϕ(z), and this completes the proof of the proposition. 2 Lemma 7. If space E is strictly symmetric, then ϕ(y) = ϕ(x) for every y ∈ Q (x). If, in addition, E = E(0, ∞), then ϕfin (y) = ϕfin (x) for every y ∈ Q (x). If E  L1 , then ϕcut (y) = ϕcut (x) for every y ∈ Q (x). Proof. Let z=

s

λi xi ,

i=1

 where λi  0, si=1 λi = 1, xi  0 and xi∗ = x. By Lemma 5, we obtain ϕ(z) = ϕ(x). However, y ∈ Q (x) can be approximated by such z. Since ϕ is continuous in strictly symmetric spaces, the lemma follows readily. The proofs are the same in cases of ϕfin and ϕcut . 2

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If A is a convex set, then function θ : A → R is called additive homogeneous if and only if   θ αy1 + βy2 = αθ (y1 ) + βθ (y2 ),

y1 , y2 ∈ A, α, β ∈ R+ .

Proposition 8. Let E be a strictly symmetric space and x ∈ E. Then the following assertions hold. (i) If E = E(0, 1), then ϕ is additive homogeneous on Q+ (x). (ii) If E = E(0, ∞), then ϕfin is additive homogeneous on Q+ (x). (iii) If E  L1 , then ϕcut is additive homogeneous on Q+ (x). Proof. We will only prove the first assertion. The proofs of the other two assertions are exactly the same. Let y ∈ Conv(extr(Ω+ (x))), so that y=

m

λi xi ,

i=1

 m ∗ ∗ ∗ where λi  0, m i=1 λi = 1, xi  0 and xi = x χ[0,βi ] . Denote z = i=1 λi x χ[0,βi ] and u =  ∗ βi >0 λi x χ[0,1] . By Lemma 5, ϕ(y) = ϕ(z). Since |z − u| ∈ L∞ , then ϕ(|u − z|) = 0 by Lemma 4. By the triangle inequality,     ϕ(u)  ϕ(z) + ϕ |u − z| = ϕ(z)  ϕ(u) + ϕ |u − z| = ϕ(u).  Hence, ϕ(y) = ϕ(u) = ( βi >0 λi )ϕ(x). It is clear that the last expression is additive homogeneous on the set Conv(extr(Ω+ (x))). By Lemma 3, the functional ϕ is continuous on Q+ (x). Hence, it is additive homogeneous on the set Q+ (x). 2 Proposition 9. Let E = E(0, ∞) be a symmetric space on semi-axis equipped with a Fatou norm. Suppose that E  L1 and x ∈ E. If Ω+ (x) = Q+ (x), then ϕ is additive homogeneous on Ω+ (x). Proof. It follows from Proposition 8 that ϕcut is additive homogeneous on Q+ (x). By assumption, Ω+ (x) = Q+ (x). Hence, ϕcut is additive homogeneous on Ω+ (x). It follows now from Proposition 6(iv) that ϕcut (x) = 0. This assertion and Lemma 2 imply that ϕ(x ∗ χ[0,β] ) = 0 for every finite β. Let y ∈ Conv(extr(Ω+ (x))). Hence, y=

m

λi xi ,

i=1

where λi  0,

m

i=1 λi

= 1, xi  0 and xi∗ = x ∗ χ[0,βi ] . By convexity of ϕ, ϕ(y)  ϕ

βi ∈[0,∞)

 λi xi + ϕ

βi =∞

 λi xi .

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205

However, 0ϕ

 λi xi 

βi ∈[0,∞)



λi ϕ(x ∗ χ[0,βi ] ) = 0.

βi ∈[0,∞)

It then follows that ϕ(y)  ϕ



 λi xi .

βi =∞

The converse inequality is obvious. By Lemma 5, ϕ(y) = ϕ





  λi xi = ϕ λi xi∗ = λi ϕ(x).

βi =∞

βi =∞

βi =∞

Clearly, the last expression is additive homogeneous on Conv(extr(Ω+ (x))). Hence, the functional ϕ is additive homogeneous on Q+ (x) = Ω+ (x). 2 Lemma 10. Let E = E(0, ∞) be a strictly symmetric space on (0, ∞) and x ∈ E. Suppose, that E  L1 . If P (x|A) ∈ Q (x) for every A, then ϕcut (x) = 0. Proof. Suppose that x = x ∗ . Set A = {[0, 1]} and y = P (x|A) ∈ E ∩ L∞ . By the assumption, y ∈ Q (x). By Lemmas 7 and 4, ϕcut (x) = ϕcut (y) = 0. 2 Lemma 11. Let E and x be as in Lemma 10. If L∞ ⊆ E, then ϕ(x) = 0. Proof. Due to the choice of E, we have 1 ∈ E. However, στ (1) = 1 implies ϕ(1) = 0. Thus, for every z ∈ E ∩ L∞ , we have ϕ(z) = 0. However, for every x ∈ E, we have ϕ(x ∗ χ[0,1] ) = 0 due to Lemma 10. Hence, 0  ϕ(x) = ϕ(x ∗ )  ϕ(x ∗ χ[0,1] ) + ϕ(x ∗ χ[1,∞) ) = 0 + 0 = 0. Lemma 12. Let E and x be as in Lemma 10. If y ∈ E ∩ L∞ and if t ω(y) := lim sup 0t t→∞

0

y ∗ (s) ds x ∗ (s) ds

,

then ϕ(y) = ω(y)ϕ(x). In particular, if in addition ϕ(x) > 0, then ω(y) < ∞. Proof. Fix ε > 0. There exists T > 0, such that for every t > T , t 0

  y (s)  ω(y) + ε ∗

t 0

x ∗ (s) ds.

2

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It then follows that y ≺≺ (ω(y) + ε)(x ∗ + Cχ[0,T ] ) for some constant C. By Lemma 3(iii), ϕ(y)  (ω(y) + ε)ϕ(x ∗ + Cχ[0,T ] ). By Lemma 4, ϕ(Cχ[0,T ] ) = 0 and, therefore, ϕ(x ∗ + Cχ[0,T ] ) = ϕ(x). Hence ϕ(y)  ω(y)ϕ(x). Now, fix ω < ω(y). There exists a sequence tk → ∞, such that tk

tk

∗

y (s) ds  ω 0

x ∗ (s) ds.

0

Without loss of generality, t0 = 0. Set u = P (x ∗ |A), where A = {[tk , tk+1 )}. It then follows that ωu ≺≺ y and ωϕ(u)  ϕ(y). However, u ∈ Q (x) and ϕ(u) = ϕ(x) due to Lemma 7. Hence ω(y)ϕ(x)  ϕ(y). 2 Proposition 13. Let E = E(0, ∞) be a symmetric space on the semi-axis and let x ∈ E. If ϕ(x) = 0, then xχA ∈ Q (x) for every Lebesgue measurable subset A ⊆ (0, ∞). Proof. Let [0, ∞) = B ∪ C,  where B, C are disjoint sets such that m(B) = m(A) and m(C) = ∞. Fix a partition C = n+1 i=1 Ci , where m(Ci ) = m(R+ \ A), 1  i  n. Let γ : B → A and γi : Ci → R+ \ A, 1  i  n be measure-preserving transformations. Define functions xni , 1  i  n by the following construction. Set xni χB = x ◦ γ , xni |Ci = x ◦ γi and xni |Cj = 0 if i = j . Clearly, xni ∼ x and n 1 i 1 1 xn = σn (xχ[0,∞)\A ) E  σn (x ∗ ) E → 0. (xχA ) ◦ γ − n n n i=1

E

Hence, (xχA ) ◦ γ ∈ Q (x). Thus, xχA ∈ Q (x).

2

Corollary 14. Let E = E(0, ∞) be a symmetric space on semi-axis. If ϕ(x) = 0, then yχA ∈ Q (x) for every y ∈ Q (x). Proof. It follows from assumption and Lemma 7 that ϕ(y) = ϕ(x) = 0. Lemma 13 implies that yχA ∈ Q (y). Since yχA ∈ Q (y) and y ∈ Q (x), then Lemma 16 implies yχA ∈ Q (x). 2 An assertion somewhat similar to the lemma below is contained in [3, Lemma 1.3]. Lemma 15. Assume that x ∈ E satisfies conditions of Proposition 13. If y ∈ Q (x) and 0  z  y, then z ∈ Q (x). Proof. Define sets eni , i = 1, . . . , n by 

i i −1 y(t)  z(t)  y(t) . eni = t: n n  Define functions ynk , k = 1, . . . , n as ynk = y k<(i+n)/2 χeni . By Corollary 14, ynk ∈ Q (x). Put 1 k yn ∈ Q (x). n n

sn =

k=1

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207

Clearly,     sn (t) − y(t) + z(t) /2  2y(t) , n

∀t ∈ eni .

Hence, sn → (y + z)/2 by norm. Therefore, (y + z)/2 ∈ Q (x). We can repeat this procedure n times and obtain 2−n ((2n − 1)z + y) ∈ Q (x). Therefore, z ∈ Q (x). 2 The following assertion seems to be known. We include the details of the proof for lack of a convenient reference. Lemma 16. Let E be a symmetric space either on (0, 1) or (0, ∞) and x ∈ E. If y ∈ Q (z) and z ∈ Q (x), then y ∈ Q (x). Proof. Without loss of generality, y = y ∗ , z = z∗ and x = x ∗ . Let y ∈ Q (z). Hence, for every ε> 0, one can find n ∈ N , λi ∈ R+ and measurable functions zi ∼ z, i = 1, . . . , n, such that n i=1 λi = 1 and n λi zi  ε. y − i=1

E

One can find measure-preserving transformations γi , such that zi − z ◦ γi L1 ∩L∞  ε. Hence, n λi z ◦ γi  2ε. y − i=1

E

However, z ∈ Q (x). Consequently, arguing in a similar way,one can find m ∈ N , μj ∈ R+ and measure preserving transformations δj , 1  j  n such that m j =1 μj = 1 and m μj x ◦ δj  2ε. z − j =1

E

Therefore, n m λi μj x ◦ γi ◦ δj  4ε y − i=1 j =1

and this suffices to complete the proof.

2

Remark 17. The collection of sets {Q(x), x ∈ E} also satisfies the transitivity property expressed in Lemma 16. We do not know whether this is the case for the collection {Q+ (x), x ∈ E}.

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4. Main results The implication (ii) ⇒ (i) in the following theorem is almost verbatim repetition of the argument given in [3, Lemma 3.1] for the case of finite measure. For convenience of the reader, we present here a proof of the most important case. Theorem 18. (a) Let E be a fully symmetric space and x ∈ E. If E = E(0, 1) or E = E(0, ∞) and E  L1 , then the following conditions are equivalent. (i) P (x|A) ∈ Q (x) for every A ∈ A. (ii) ϕ(x) = 0. (b) If E = E(0, ∞) and E ⊆ L1 , then the following conditions are equivalent. (i) P (x|A) ∈ Q (x) for every A ∈ A. (ii) ϕfin (x) = 0. Proof. (a) (i) ⇒ (ii) Let E = E(0, 1) and x = x ∗ . Set A = {[0, 1]} and y = P (x|A). By assumption, y ∈ Q (x). By Lemmas 7 and 4, ϕ(x) = ϕ(y) = 0. Let E = E(0, ∞) and L∞ ⊆ E  L1 . The assertion is proved in Lemma 11. Let E = E(0, ∞) and L∞  E  L1 . Suppose that x = x ∗ and ϕ(x) > 0. Set B = {[0, 1]}, t  ψ = P (x|B) and ψ(t) = 0 ψ  (s) ds. By Lemma 7, ϕ(ψ  ) = ϕ(x). Let y ∈ E ∩ L∞ . It follows from Lemma 12, that ω(y) < ∞. Therefore, y ∈ Mψ . Hence, E ∩ L∞ ⊆ Mψ . Since E is fully symmetric and ψ  ∈ E ∩ L∞ , then Mψ ⊆ E ∩ L∞ . Therefore, E ∩ L∞ = M ψ . If u = 2σ 1 ψ  , then ϕ(u) = ϕ(ψ  ) by Lemma 3(v). Hence ω(u)ϕ(x) = ϕ(x) and ω(u) = 1. 2 However, t 2x(2s) ds ψ(2t) . = lim sup ω(u) = lim sup 0 t t→∞ t→∞ ψ(t) 0 x(s) ds Thus, lim

t→∞

ψ(2t) = 1. ψ(t)

(14)

Let G be the set defined by

 y ∗ (t) G = y ∈ E: ∃C sup  <∞ . t1 ψ (Ct) Note, that our set G differs from the one introduced in [10]. If y1 , y2 ∈ G, then yi∗ (t)  Ci ψ(Ct) for t  12 . It then follows (y1 + y2 )∗ (t)  y1∗





 t t C t . + y2∗  (C1 + C2 )ψ  2 2 2

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209

In particular, G is a linear set and Conv({y ∗ = x ∗ }) ⊆ G. If the condition (14) holds, then there exists a sequence tk , such that t0 = 0, t1 = 1 and for every k ψ(tk+1 ) − ψ(tk ) 2 ψ( 12 tk+1 )  . tk+1 − tk 3 tk+1 Set A = {[tk , tk+1 ]} and z = P (x|A). It follows from the construction given in [10] that (z − y)χ[ 1 tk ,tk ] Mψ  14 for every y ∈ G and every sufficiently large k. However, 2

(y − z)χ[ 1 tk ,tk ] L∞ → 0. Since Mψ = E ∩ L∞ , then (z − y)χ[ 1 tk ,tk ] E  2

2

1 4

for sufficiently

(z, Q (x))

large k. In particular, y − zE  Hence, distE (z, G)  and distE  14 . This  contradicts the assumption that P (x|A) ∈ Q (x). (a) (ii) ⇒ (i) Let E = E(0, 1) or E = E(0, ∞)  L1 . We will prove the assertion for the case when A = {[0, 1]}. The general proof is similar. Without loss of generality, x decreases on [0, 1]. Define functions xni , i = 0, . . . , n − 1 such that (i) xni = x outside (0, 1) and (ii) xni (t) = x((t + ni )(mod 1)) if t ∈ (0, 1). Set xn (t) = x(t − ni ) if ni  t  i+1 n , 0  i  n − 1 and xn (t) = 0 if t  1. Clearly, xni ∼ x and (xn )∗  σn (x ∗ ). We will show that 1 4.

1

1 4

1 x(s) ds − x n n−1



i=0

0

1

  n i (mod 1)  x(s) ds t+ n 0

and 1

1 x(s) ds − x n n−1 i=0

0

  i 1 t+ (mod 1)  − xn (t). n n

We will prove only the first inequality. The proof of the second one is identical. Without loss of generality, t ∈ [0, n1 ]. Clearly, i+2

 n 1 i x t+  x(s) ds n n i+1 n

for i = 0, . . . , n − 2. Hence, 1 0

1

 n

 n−1 n−1 1 i 1 = x(s) ds − x t + x(s) ds − x t+ n n n n i=0

0 i+2   n1   n n−2 1 i x t+ − − x(s) ds  x(s) ds. n n

i=0

i+1 n

0

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Therefore,  1  n−1  1 i  1  xn (t)  zn (t),  x(s) ds −   n n

t ∈ [0, 1],

i=0

0

1 where zn = xn + ( 0 xn (s) ds)χ[0,1] . Obviously, zn ≺≺ 2xn  2σn (x ∗ ) and, therefore, zn E  2σn (x ∗ )E . It then follows that n−1 1 i 2 xn  σn (x ∗ ) E → 0. P (x|A) − n n i=0

E

(b) (i) ⇒ (ii) Let E = E(0, ∞) and E ⊂ L1 . Set A = {s: x(s)  1} and A = {A}. Set y = P (x|A) ∈ E ∩ L∞ . Lemma 4 implies that ϕ(y) = 0. By the assumption, y ∈ Q (x). By Lemma 7, ϕ(x) = ϕ(y) = 0. (b) (ii) ⇒ (i) The assertion follows from Theorem 23. 2 The following proposition is the core technical result of the article. In case of the interval (0, 1) it may be found in [3, Lemma 3.2]. However, our proof is more general, simpler and shorter. We consider the functions of the form x=



xi χ[ai−1 ,ai ] ,



y=

i∈Z

yi χ[ai−1 ,ai ] ,

(15)

i∈Z

where {ai }i∈Z is an increasing sequence (possibly finite or one-sidedly infinite). Proposition 19. Let y = y ∗ and x = x ∗ be the functions of the form (15) either on (0, 1) or on (0, ∞). If y ≺≺ x, then there exists a countable collection {k }k∈K of disjoint sets, where k = Ik ∪ Jk with intervals Ik and Jk of finite measure, such that (i) the functions x and y are constant on the intervals Ik and Jk and the interval Ik lies to the left of Jk , k ∈ K. (ii) y|k ≺ x|k , k ∈ K.  (iii) y(t)  x(t) if t ∈ / k∈K k . If, in addition, x and y are functions on (0, 1) and t∈ / k∈K k .

1 0

y(s) ds =

1 0

x(s) ds, then y(t) = x(t) if

Proof. There  exists a subsequence {ami }i∈I (possibly finite or one-sidedly infinite) such that {x < y} = ∈I [ami −1 , ami ]. Since y ≺≺ x, we have t

t (x − y)+ (s) ds −

0

t (y − x)+ (s) ds =

0

t x(s) ds −

0

y(s) ds  0. 0

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211

For each i ∈ I, denote by bi the minimal t > 0, such that t

ami (x − y)+ (s) ds = (y − x)+ (s) ds.

0

0

Clearly, for every i ∈ I, ami −1



ami ami (x − y)+ (s) ds = (x − y)+ (s) ds  (y − x)+ (s) ds.

0

0

0

Hence, bi  ami −1 . For each i ∈ I, the set [bi−1 , bi ] ∩ {x > y} is a finite union disjoint intervals on which each of x and y is finite. By the definition of bi , we have ami

bi (y − x)+ (s) ds =

ami −1

(x − y)+ (s) ds =

ni  j =1

bi−1

ni

j j =1 Ii

of

(x − y)+ (s) ds.

j

Ii

j

Set K = {(i, j ): 1  j  ni , i ∈ I}. If k = (i, j ) ∈ K, set Ik = Ii and  j j −1 j Jk = Ji = ami −1 + (ymi − xmi )−1 ci , ami −1 + (ymi − xmi )−1 ci , where j ci

=

j  l=1

(x − y)+ (s) ds,

i ∈ I, 0  j  ni .

Iil

Using the fact that x and y are constant on the interval [ami −1 , ami ], we obtain Jk ⊂ [ami −1 , ami ]  i j Ji = [ami −1 , ami ]. and nj =1 (i) Both x and y are constant on Ik and Jk , k ∈ K. Since bi  ami −1 for each i ∈ I, then Ik lies to the left of Jk for k ∈ K. It then follows from (i), that 

 (x − y)+ (s) ds =

Ik

(y − x)+ (s) ds,

k ∈ K.

(16)

Jk

(ii) Since x|Ik  y|Ik and x|Jk  y|Jk for all k ∈ K, then the assertion follows directly from (i) and (16).  ni  j (iii) The set {y > x} = i∈I j =1 Ji ⊆ k∈K k . The last assertion is immediate. 2

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Corollary 20. Let E be a fully symmetric space either on the interval (0, 1) or on the semi-axis. If x, y and B = {k }k∈K are as in Proposition 19 and y(t) = x(t) if t ∈ / k k , then y can be arbitrary well approximated in the norm of E by convex combinations of functions of the form P (x|A), A ∈ A. Proof. Set λk = (y|Ik − y|Jk )/(x|Ik − x|Jk ), k ∈ K. Since y|k ≺ x|k , it is not difficult to verify that λk ∈ [0, 1], k ∈ K. Further, a simple calculation shows that y = (1−λk )P (x|B)+λk x on k , k ∈ K. As is well known, every [0, 1]-valued sequence can be uniformly approximated by convex combinations of {0, 1}-valued sequences. n Fix ε > 0. There nexists μ ∈ l∞ (K) with μ = i=1 θi χDi for some n ∈ N , 0  θi ∈ R and Di ⊆ K such that i=1 θi = 1 and λ − μ∞  ε. Set z = (1 − μk )P (x|B) + μk x on k , k ∈ K and z = x  outside k∈K k . It is clear that |y − z|χk = |λk − μk ||x − P (x|B)|χk , k ∈ K and |y − z| =  k∈K |y − z|χk  2ε(x + P (x|B)). Therefore, y − zE  2εxE . Set Fi = k∈Di k and Ai = {k }k ∈D / i ∈ A, 1  i  n. It is then clear that z=

n n   θi (1 − χFi )P (x|B) + χFi x = θi P (x|Ai ). i=1

2

i=1

4.1. The case that E ⊆ L1 Theorem 21. Let E = E(0, 1) be a fully symmetric space on the interval (0, 1). If x ∈ E, then the following statements are equivalent. (i) Ω  (x) = Q (x). (ii) ϕ(x) = 0. Proof. (i) ⇒ (ii) Suppose that Q (x) = Ω  (x). Set A = {[0, 1]} and y = P (x|A). Clearly, y ∈ Ω  (x) = Q (x). Lemma 7 implies that ϕ(x) = ϕ(y). Lemma 4 implies ϕ(y) = 0. The assertion is proved. (ii) ⇒ (i) Let x = x ∗ and 0  y ∈ Ω  (x). In this case, y = y ∗ ◦ γ for some measure-preserving transformation γ (see [15] or [2, Theorem 7.5, p. 82]). Without loss of generality, we may assume that y = y ∗ . Fix ε > 0. Set   sn (ε) = inf s: y(s)  y(1) + nε ,

n ∈ N.

Let Aε be the partition, determined by the points sn (ε), n ∈ N . Set u = P (y|Aε ) and z = P (x|Aε ). The functions u and z satisfy the condition u ≺ z and are of the form given in (15). By Lemma 3(iii), ϕ(z)  ϕ(x) = 0. By Theorem 18, P (z|A) ∈ Q (z) for every A ∈ A. It follows now from Corollary 20 that u ∈ Q (z). However, z ∈ Q (x) by Theorem 18. Therefore, by Lemma 16, u ∈ Q (x). However, y − uL∞  ε. Since ε is arbitrary, y ∈ Q (x). 2 Theorem 22. Let E = E(0, 1) be a fully symmetric space on the interval (0, 1). If x ∈ E and ϕ(x) = 0, then Ω+ (x) = Q+ (x). If, in addition, the norm on E is a Fatou norm, then converse assertion also holds.

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213

Proof. Suppose that ϕ(x) = 0 and let y ∈ Ω+ (x). Hence, there exists s0 ∈ [0, 1], such that  s0 ∗ 1 ∗ ∗   0 x (s) ds = 0 y (s) ds. Set z = x χ[0,s0 ] . By Theorem 21, y ∈ Q (z). Hence, y ∈ Q (z) ⊆ Q+ (x). By Proposition 6, there exist 0  y, z ∈ E, such that x = y + z and ϕ(x) = ϕ(y) = ϕ(z). By Proposition 8, ϕ(x) = ϕ(y) + ϕ(z). Consequently, ϕ(x) = 0. 2 Now, consider the case that E = E(0, ∞). Theorem 23. Let E = E(0, ∞) be a fully symmetric space on semi-axis. If E ⊆ L1 and x ∈ E, then the following assertions are equivalent. (i) Ω  (x) = Q (x). (ii) ϕfin (x) = 0. Proof. (i) ⇒ (ii) Let x = x ∗ and suppose that Q (x) = Ω  (x). Set A = {[0, 1]} and y = P (x|A). Clearly, y ∈ Ω  (x) = Q (x). Lemma 7 implies that ϕfin (x) = ϕfin (y). Lemma 4 implies ϕfin (y) = 0. The assertion is proved. (ii) ⇒ (i) Let x = x ∗ and 0  y ∈ Ω  (x). It follows from [10, Lemma II.2.1] that for every fixed ε > 0 there exists a measure-preserving transformation γ such that y − y ∗ ◦ γ E  ε. Without loss of generality, we may assume that y = y ∗ . For every S > 0, 1 (στ x)χ[0,S]  S (στ x)χ[0,1] → 0. E E τ τ (a) Suppose first that supp(x) = supp(y) = (0, ∞). Fix ε > 0. There exists T , such that xχ[T ,∞) L1 ∩L∞  ε,

yχ[T ,∞) L1 ∩L∞  ε.

∞ S T T Clearly, 0 x(s) ds < 0 x(s) ds. Hence, there exists S  T , such that 0 y(s) ds = 0 x(s) ds. By Theorem 21, yχ[0,S] ∈ Q (xχ[0,T ] ). Hence, y ∈ Q (x) + yχ(S,∞) − Q (xχ(T ,∞) ) and, therefore, dist(y, Q (x))  2ε. Since ε is arbitrary, y ∈ Q (x). (b) Suppose now that m(supp(x)) < ∞ or m(supp(y)) < ∞. Fix z = z∗ ∈ L1 ∩ L∞ with infinite support. It is clear that (y + εz) ∈ Ω  (x + εz), ε > 0. By assumption and Lemma 4, ϕfin (x + εz) = 0. Hence, using (a) preceding, it follows that (y + εz) ∈ Q (x + εz) ⊂ Q (x) + εQ (z). Hence, dist(y, Q (x))  ε for every ε > 0 and, therefore, y ∈ Q (x). 2 Theorem 24. Let E = E(0, ∞) be a fully symmetric space on (0, ∞) such that E ⊆ L1 . If 0  x ∈ E and ϕfin (x) = 0, then Ω+ (x) = Q+ (x). If, in addition, the norm on E is a Fatou norm, then converse assertion also holds. Proof. Let ϕfin (x) = 0 and y ∈ Ω+ (x). As in Theorem 23, we may assume y = y ∗ . Fix ε > 0. There exists T > 0 such that xχ[T ,∞) L1 ∩L∞  ε,

yχ[T ,∞) L1 ∩L∞  ε.

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Select S  T such that S

∗

T

x (s) ds = 0

y ∗ (s) ds.

0

Clearly, yχ[0,T ] ∈ Ω  (x ∗ χ[0,S] ). By Theorem 21, yχ[0,T ] ∈ Q (x ∗ χ[0,S] ) ⊆ Q+ (x). Hence, y ∈ Q+ (x). By Proposition 6, there exist 0  y, z ∈ E, such that x = y + z and ϕfin (x) = ϕfin (y) = ϕfin (z). By Proposition 8, ϕfin (x) = ϕfin (y) + ϕfin (z). Consequently, ϕfin (x) = 0. 2 4.2. The case that E  L1 Theorem 25. Let E = E(0, ∞) be a fully symmetric space on the semi-axis and let x ∈ E. If ϕ(x) = 0, then Ω+ (x) = Q (x). Proof. Let us assume first that y = y ∗ ∈ Ω+ (x). Fix ε > 0. Set tn (ε) = 1 + nε,   sn (ε) = inf s: y(s)  y(1) + nε ,   s−n (ε) = sup s: y(s)  y(1) − nε . Let Aε be the partition, determined by the points s±n (ε), tn (ε). Set u = P (y|Aε ) and z = P (x|Aε ). The functions u and z satisfy the conditions u ≺≺ z and (15). Set v=u

k∈K

χk + zχ(0,∞)\k∈K k ,

where the collection {k }k∈K is given by Proposition 19. By Lemma 3(iii), ϕ(z)  ϕ(x) = 0. By Theorem 18, P (z|A) ∈ Q (z) for every A ∈ A. It follows now from Corollary 20 that v ∈ Q (z). Since u  v, it follows from Lemma 15 that u ∈ Q (z). Theorem 18 implies that z ∈ Q (x). By Lemma 16, u ∈ Q (x). However,   dist y, Q (x)  y − uE  y − P (y|Aε ) L

1 ∩L∞

   ε 1 + y(1) .

Since ε is arbitrary, y ∈ Q (x). Let now y ∈ Ω+ (x) be arbitrary. By [10, Lemma II.2.1 and Theorem II.2.1], for every fixed ε > 0, there exist y1 ∈ E, y2 ∈ E, y = y1 + y2 and measure-preserving transformation γ such that 0  y1  y ∗ ◦ γ and y2 E  ε. Since we already proved that y ∗ ∈ Q (x), the assertion follows immediately. 2 Theorem 26. Let E = E(0, ∞) be a fully symmetric space on semi-axis. Suppose that E  L1 and x ∈ E. If ϕ(x) = 0, then the set Ω+ (x) is the norm-closed convex hull of its extreme points. If, in addition, the norm on E is a Fatou norm, then converse assertion also holds. Proof. Suppose that ϕ(x) = 0. Applying Theorem 25 and noting the embedding Q (x) ⊆ Q+ (x) yields the assertion.

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215

Conversely, by Proposition 6(iv), there exist 0  y1 , z1 ∈ E, such that x = y1 + z1 and ϕcut (x) = ϕcut (y1 ) = ϕcut (z1 ). By assumption, Ω+ (x) = Q+ (x) and so y1 , z1 ∈ Q+ (x). By Proposition 8, ϕcut (x) = ϕcut (y1 ) + ϕcut (z1 ). Consequently, ϕcut (x) = 0. By Proposition 6(ii), there exist 0  y2 , z2 ∈ E, such that x = y2 + z2 and ϕ(x) = ϕ(y2 ) = ϕ(z2 ). Again, by the assumption, we have y2 , z2 ∈ Q+ (x) and therefore, by Proposition 9, we have ϕ(x) = ϕ(y2 ) + ϕ(z2 ). Consequently, ϕ(x) = 0. 2 Acknowledgments We would like to thank Peter Dodds for many helpful comments on the content of this paper and lengthy discussions of earlier drafts. We also thank Sergei Astashkin, Aleksandr Sedaev and Evgenii Semenov for their interest. Appendix A A.1. An application to the case of orbits Ω(x) The following consequence of Theorem 22 is essentially due to Braverman and Mekler [3]. Corollary 27. If ϕ(x) = 0, then Ω(x) is the norm-closed convex hull of its extreme points. Proof. Let x = x ∗ and y ∈ Ω(x). Clearly, y = u · |y|, where |u| = 1 a.e. and |y| ∈ Ω+ (x). Fix ε > 0. By  Theorem 22, there exist n ∈ N , scalars λn,i , βn,i ∈ [0, 1] and functions xn,i ∼ xχ[0,βn,i ] , such that ni=1 λn,i = 1 and n λn,i xn,i  ε. |y| − i=1

E

There exist measure-preserving transformations γn,i , 1  i  n (see [15]), such that xn,i = 1 =u·x ◦γ 2 (x ∗ χ[0,βn,i ] ) ◦ γn,i . Set xn,i n,i and xn,i = u · (xχ[0,βn,i ] − xχ[βn,i ,1] ) ◦ γn,i , 1  i  n. It is clear that xn,i ∼ x, 1  i  n, and n n 1 1 1 2 λn,i xn,i − λn,i xn,i  ε. 2 y − 2 2 i=1

i=1

E

A.2. Extreme points of the orbit Ω+ (x) The following theorem is due to Ryff (see [14]). Theorem 28. If 0  x ∈ L1 (0, 1), then y ∈ extr(Ω  (x)) if and only if y ∗ = x ∗ . Corollary 29. If 0  x ∈ L1 (0, 1), then y ∈ extr(Ω+ (x)) if and only if y ∗ = x ∗ χ[0,β] for some β  0. 1 β Proof. Indeed, if 0 x ∗ (s) ds = 0 y ∗ (s) ds, then y ∈ Ω  (x ∗ χ[0,β] ). Therefore, if y ∈ extr(Ω+ (x)), then obviously y ∈ extr(Ω  (x ∗ χ[0,β] )) and the assertion follows immediately from Theorem 28.

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t t If y ∗ = x ∗ χ[0,β] and y = 12 (u1 + u2 ) with ui ∈ Ω+ (x), then 0 u∗i (s) ds = 0 x ∗ (s) ds for t ∈ [0, β] and supp(ui ) = supp(y). Therefore, (u1 + u2 )∗ = u∗1 + u∗2 . It follows now from [10, (II.2.19)] that u1 = u2 . 2 Lemma 30. If 0  x ∈ L1 + L∞ and y ∈ extr(Ω+ (x)), then yχ{y 0, then either y ∗ = x ∗ χ[0,β] for some β ∈ [0, ∞) or y ∗ = x ∗ and yχ{y 0 and find t2  t1  t1 + t2 ∗ ∗ such that 0 x (s) ds = 0 y (s) ds. Clearly, y ∗ χ[0,t1 ] ≺ x ∗ χ[0,t2 ] and y ∗ χ[t1 ,∞) ≺≺ x ∗ χ[t2 ,∞) . If y ∗ χ[0,t1 ] = 12 (u1 + u2 ) with u1 , u2 ∈ Ω  (x ∗ χ[0,t2 ] ), then set yi = ui χ[0,t1 ] + y ∗ χ[t1 ,∞) . We claim yi ≺≺ x. Indeed, if e ∈ (0, ∞) and m(e) < ∞, then e = e1 ∪ e2 with e1 ⊂ [0, t1 ] and e2 ⊂ [t1 , ∞). Therefore, 

 yi (s) ds = e

 ui (s) ds +

e1



u∗i (s) ds

y (s) ds  e2

min{t2 ,m(e1 )}

∗

t1 +m(e  2)

m(e  1)

∗

y ∗ (s) ds

+ t1

0 t2 +m(e  2)

x (s) ds + 0

∗

m(e) 

x ∗ (s) ds.

x (s) ds  t2

0

/ extr(Ω+ (x)). Therefore, y ∗ χ[0,t1 ] ∈ Hence, yi ∈ Ω+ (x) and y = 12 (y1 + y2 ). Thus, y ∈  ∗ ∗ ∗ extr(Ω (x χ[0,t2 ] )). By Theorem 28, y = x on [0, t2 ]. The assertion follows now from Lemma 30. The converse assertion is easy. 2 Corollary 32. If x ∈ L1 (0, ∞), then 0  y ∈ extr(Ω  (x)) if and only if y ∗ = x ∗ . The proof is identical to that of Corollary 31. A.3. Marcinkiewicz spaces with trivial functional ϕ It follows from Lemma 3 and the definition of Marcinkiewicz space, that ϕ = 0 if and only if ϕ(ψ  ) = 0. It is now easy to derive, that in case of the interval (0, 1) this is equivalent to the

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217

condition lim inf

ψ(2t) > 1. ψ(t)

lim inf

ψ(2t) >1 ψ(t)

t→0

In case of the semi-axis, the condition

t→∞

needs to be added. A.4. A comparison of conditions (1) and (2) in Orlicz spaces Let M be a convex function satisfying (5) and let LM be the corresponding Orlicz space on (0, 1). The following proposition shows that LM always satisfies condition (2). Proposition 33. We have ϕ(x) = 0 for every x ∈ LM . Proof. Using the description of relatively weakly compact subsets in LM given in [1] (see also [12, p. 144]) we see that for every 0  y ∈ LM 1

n n

M

 1 y(s) ds → 0. n

0

We are going to prove that n1 σn xLM → 0 for every x ∈ LM . Assume the contrary. Let σn xLM  nα for some 0  x ∈ LM , some α > 0 and for arbitrary large n  1. By the definition of the norm  · LM , we have

1 M

 1 σn x(s) ds  1. nα

0

Hence, 1

n n

 1 y(s) ds  1 M n

0

with y = α −1 x ∈ LM . A contradiction.

2

However, there exists an Orlicz space LM which fails to satisfy condition (1). For the definition of Boyd indices 1  pE  qE  ∞ of a fully symmetric space E, we refer the reader to [11, 2.b.1 and p. 132]. It is clear, that the condition (1) holds for a fully symmetric space E if and only if pE > 1. However (see e.g. [11]) Orlicz space LM is separable if and only if qLM < ∞. It is well known that there exists a non-separable Orlicz space LM with pLM = 1.

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A.5. An application to symmetric functionals Let E be a fully symmetric space. A positive functional f ∈ E ∗ is said to be symmetric (respectively, fully symmetric) if f (y) = f (x) (respectively, f (y)  f (x)) for all 0  x, y ∈ E such that y ∗ = x ∗ (respectively, y ≺≺ x). We refer to [8,5] and references therein for the exposition of the theory of singular fully symmetric functionals and their applications. Recently, symmetric functionals which fail to be fully symmetric were constructed in [9] on some Marcinkiewicz spaces. However, for Orlicz spaces situation is different. The following proposition shows that a symmetric functional on an Orlicz space on the interval (0, 1) is necessary fully symmetric. Proposition 34. Any symmetric functional on LM is fully symmetric. Proof. Let ω ∈ E ∗ be symmetric. It is clear, that ω(x ∗ χ[0,β] )  ω(x) for x  0. Therefore, ω(y)  ω(x) for y ∈ Conv{y ∗ = x ∗ χ[0,β] }. Since ω is continuous, we have ω(y)  ω(x) for y ∈ Q+ (x). By Theorem 22 and Proposition 33, we have Q+ (x) = Ω+ (x), and so ω is a fully symmetric functional on LM . 2 Corollary 35. Any singular symmetric functional on LM vanishes. Proof. Indeed, there are no fully symmetric singular functionals on LM (see [8, Theorem 3.1]). 2 We also formulate the following hypothesis: If E is a fully symmetric space, then functional ϕ vanishes if and only if there are no singular symmetric functionals on E. References [1] T. Ando, Weakly compact sets in Orlicz spaces, Canad. J. Math. 14 (1962) 170–176. [2] C. Bennett, R. Sharpley, Interpolation of Operators, Pure Appl. Math., vol. 129, Academic Press Inc., Boston, MA, 1988. [3] M.Sh. Braverman, A.A. Mekler, The Hardy–Littlewood property for symmetric spaces, Siberian Math. J. 18 (1977) 371–385. [4] A.-P. Calderón, Spaces between L1 and L∞ and the theorem of Marcinkiewicz, Studia Math. 26 (1966) 273–299. [5] A.L. Carey, F.A. Sukochev, Dixmier traces and their applications in noncommutative geometry, Russian Math. Surveys 61 (2006) 45–110. [6] V.I. Chilin, A.V. Krygin, F.A. Sukochev, Extreme points of convex fully symmetric sets of measurable operators, Integral Equations Operator Theory 15 (1992) 186–226. [7] P.G. Dodds, F.A. Sukochev, G. Schlüchtermann, Weak compactness criteria in symmetric spaces of measurable operators, Math. Proc. Cambridge Philos. Soc. 131 (2001) 363. [8] P.G. Dodds, B. de Pagter, E.M. Semenov, F.A. Sukochev, Symmetric functionals and singular traces, Positivity 2 (1998) 47–75. [9] N. Kalton, F. Sukochev, Rearrangement-invariant functionals with applications to traces on symmetrically normed ideals, Canad. Math. Bull. 51 (2008) 67–80. [10] S.G. Krein, Ju.I. Petunin, E.M. Semenov, Interpolation of Linear Operators, Nauka, Moscow, 1978 (in Russian); English translation in: Transl. Math. Monogr., vol. 54, Amer. Math. Soc., Providence, RI, 1982. [11] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I and II: Sequence Spaces; Function Spaces, Springer, 1996. [12] M.M. Rao, Z.D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Marcel Dekker Inc., New York, 1991. [13] J.V. Ryff, Orbits of L1 -functions under doubly stochastic transformations, Trans. Amer. Math. Soc. 117 (1965) 92–100. [14] J.V. Ryff, Extreme points of some convex subsets of L1 (0, 1), Proc. Amer. Math. Soc. 18 (1967) 1026–1034. [15] J.V. Ryff, Measure preserving transformations and rearrangements, J. Math. Anal. Appl. 31 (1970) 449–458.

Journal of Functional Analysis 257 (2009) 219–242 www.elsevier.com/locate/jfa

Some s-numbers of an integral operator of Hardy type on Lp(·) spaces D.E. Edmunds a,∗ , J. Lang b,∗ , A. Nekvinda c,1 a Department of Mathematics, Mantell Building, University of Sussex, Brighton BN1 9RF, UK b Department of Mathematics, The Ohio State University, 100 Math Tower, 231 West 18th Avenue,

Columbus, OH 43210-1174, USA c Department of Mathematics, Faculty of Civil Engineering, Czech Technical University, Thákurova 7,

16629 Prague 6, Czech Republic Received 25 October 2008; accepted 23 February 2009 Available online 6 March 2009 Communicated by C. Kenig

Abstract Let I = [a, b] ⊂ R, let p : I → (1, ∞) be either a step-function or strong log-Hölder continuous on I , let p(·) p(·) Lp(·) (I ) be the usual space of Lebesgue type with  x variable exponent p, and let T : L (I ) → L (I ) be the operator of Hardy type defined by Tf (x) = a f (t) dt. For any n ∈ N, let sn denote the nth approximation, Gelfand, Kolmogorov or Bernstein number of T . We show that    1/p(t)   1 p (t)p(t)p(t)−1 sin π/p(t) dt lim nsn = n→∞ 2π I

where p  (t) = p(t)/(p(t) − 1). The proof hinges on estimates of the norm of the embedding id of Lq(·) (I ) in Lr(·) (I ), where q, r : I → (1, ∞) are measurable, bounded away from 1 and ∞, and such that, for some ε ∈ (0, 1), r(x)  q(x)  r(x) + ε for all x ∈ I . It is shown that   min 1, |I |ε  id  ε|I | + ε−ε , a result that has independent interest. * Corresponding author.

E-mail addresses: [email protected] (D.E. Edmunds), [email protected] (J. Lang), [email protected] (A. Nekvinda). URL: http://www.math.ohio-state.edu/~lang/ (J. Lang). 1 The third author was supported by grant no. MSM 201/08/0383 of the Grant Agency of the Czech Republic. 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.02.015

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© 2009 Elsevier Inc. All rights reserved. Keywords: Hardy-type operator; Compactness; s-Numbers; Lp(·)

1. Introduction Let I = [a, b] be a compact interval in the real line and let T be the operator of Hardy type given by x Tf (x) :=

f (t) dt

(x ∈ I ).

(1.1)

a

It is well known that if p ∈ (1, ∞), then T is a compact map from Lp (I ) to Lp (I ) (see, for example, [2, Chapter 2, §3]); moreover, if sn (T ) stands for the nth approximation, Bernstein, Gelfand or Kolmogorov number of T , then lim nsn (T ) = γp (b − a)/2,

n→∞

(1.2)



where γp = π −1 p 1/p (p  )1/p sin(π/p) and p  = p/(p − 1). We refer to [2] and [7] for details of this and similar results for more general operators of Hardy type. The position when T is viewed as a map from Lp (I ) to Lq (I ) and p = q is less simple, but nevertheless genuine asymptotic results similar to (1.2) have been obtained for various s-numbers of T in particular circumstances: see [3] and [5]. The focus of the present paper is on the behavior of s-numbers of the map T when it acts from the variable exponent space Lp(·) (I ) to Lp(·) (I ). Here p : I → (1, ∞) and by Lp(·) (I ) is meant the space of all real-valued functions f on I such that for some λ > 0, 

  f (x)/λp(x) dx < ∞;

I

endowed with the norm

 p(x)  f p(·) := inf λ > 0: f (x)/λ dx  1

(1.3)

I

it is a Banach space. Because of their natural occurrence in various significant physical contexts (see [12]), these spaces (which are particular cases of Musielak–Orlicz spaces) have been intensively studied in recent years, considerable emphasis being placed on the properties on them of such classical operators in harmonic analysis as the Hardy–Littlewood maximal operator. Our main result is a direct analogue of (1.2): if p is either a step-function or a strong log-Hölder continuous function (see Definition 4.10 and note that any Lipschitz or Hölder function p(·) is a strong log-Hölder continuous function), then

D.E. Edmunds et al. / Journal of Functional Analysis 257 (2009) 219–242

1 lim nsn (T ) = n→∞ 2π



   1/p(x)  sin π/p(x) dx, p (x)p(x)p(x)−1

221

(1.4)

I

where sn (T ) is the nth approximation, Gelfand, Kolmogorov or Bernstein number of T : Lp(·) (I ) → Lp(·) (I ). So far as we are aware, this is the first result concerning the s-numbers of operators acting on spaces with variable exponent, despite the clear importance of these numbers and the considerable literature devoted to them in the context of classical Lebesgue spaces. A key step in the proof is the following two-sided estimate of the norm of the embedding id of Lq(·) (I ) in Lp(·) (I ) when, for ε ∈ (0, 1), p(x)  q(x)  p(x) + ε (x ∈ I ):   min 1, |I |ε  id  ε|I | + ε −ε , This has intrinsic interest, being a sharp improvement of the classical embedding theorem for Lp(·) spaces due to Ková˘cík and Rákosník [6]. 2. Preliminaries Throughout the paper I will stand for a compact interval [a, b] in the real line R, and given any measurable subset E of I, the Lebesgue measure of E will be denoted by |E| and the characteristic function of E by χE . By M(I ) is meant the family of all extended scalar-valued (real or complex) measurable functions on I, and P(I ) will stand for the subset of M(I ) consisting of all those functions p(·), with values in (1, ∞), such that 1 < p− := ess inf p(x)  p+ := ess sup p(x) < ∞. x∈I

x∈I

For all f ∈ M(I ), define  ρp(·) (f ) =

  f (x)p(x) dx

I

and   f p(·),I = f p(·) = inf λ > 0: ρp(·) (f/λ)  1 . The generalised Lebesgue space Lp(·) (I ) (or space with variable exponent) is the set   Lp(·) (I ) := f : f p(·) < ∞ , equipped with the norm  · p(·) ; it is routine to verify that it is a Banach space; indeed, it is a Banach function space. We refer to [6] for an account of the fundamental properties of these spaces and in particular for the following basic embedding theorem, in which by X → Y we mean that the Banach space X is continuously embedded in the Banach space Y. Theorem 2.1. Let p(·), q(·) ∈ P(I ) be such that for all x ∈ I, p(x)  q(x). Then Lq(·) (I ) → Lp(·) (I ).

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In what follows we consider the Hardy operator T acting on a space Lp(·) (I ) with variable exponent. To establish its compactness we use the following well-known result concerning its behaviour on classical Lebesgue spaces. Theorem 2.2. Let r, s ∈ (1, ∞). Then T maps Lr (I ) compactly into Ls (I ). This follows from [2], Theorems 2.3.1 and 2.3.4, for example. Now the compactness of T on spaces with variable exponent follows quickly. Lemma 2.3. Let 1 < c < d < ∞ and suppose that p(·), q(·) ∈ P(I ) are such that for all x ∈ I we have p(x), q(x) ∈ (c, d). Then T maps Lp(·) (I ) compactly into Lq(·) (I ). Proof. By Theorem 2.1, Lp(·) (I ) and Ld (I ) are continuously embedded in Lc (I ) and Lq(·) (I ), respectively. Moreover, by Theorem 2.2, T maps Lc (I ) compactly into Ld (I ). The result now follows by composition of these maps. 2 More detailed information about the compactness properties of T is provided by the approximation, Bernstein, Gelfand and Kolmogorov numbers, and we next recall the definition of these quantities. Let X and Y be Banach spaces and let S : X → Y be compact and linear. Then given any n ∈ N, the nth approximation number of S is defined to be an (S) = inf S − F , where the infimum is taken over all bounded linear maps F : X → Y with rank less than n; the nth Bernstein number of S is bn (S) = sup

inf

x∈Xn \{0}

SxY /xX ,

where the supremum is taken over all n-dimensional subspaces Xn of X; the nth Gelfand number of S is   X  : M is a linear subspace of X, codim X < n , cn (S) = inf SJM X is the embedding map from M to X; and the nth Kolmogorov number of S is where JM

dn (S) = inf

sup

inf Sf − gY /f X ,

Xn 0<f X 1 g∈Xn

where the outer infimum is taken over all n-dimensional subspaces Xn of X. Further details of these numbers and their basic properties will be found in [1,9] and [11]; for the moment we simply note that the approximation numbers are the largest of them. We recall that not all these s-numbers have the multiplicative property detailed in [1], p. 72: the Bernstein numbers fail to have it (see [10]). However, every s-number sn satisfies the following inequality sn (R ◦ T ◦ S)  Rsn (T )S, for arbitrary and appropriately composed bounded linear maps R, S and T .

(2.1)

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223

To determine the properties of T we introduce certain functions that will play a key role in our analysis. Definition 2.4. Let p(·), q(·) ∈ P(I ), let J = (c, d) ⊂ I, and let ε > 0. We define  ·      Ap(·),q(·) (J ) = inf sup  f    y∈J y

 ·      Bp(·) (J ) = inf sup  f    y∈J y

: f p(·),J  1 , q(·),J

pJ+ ,J

: f p− ,J  1 , J

  Cp(·),q(·) (J ) = sup Tf q(·),J : f p(·),J  1, (Tf )(c) = (Tf )(d) = 0 ,   Dp(·) (J ) = sup Tf p− ,J : f p+ ,J  1, (Tf )(c) = (Tf )(d) = 0 J

J

where pJ− = inf{p(x); x ∈ J } and pJ+ = sup{p(x); x ∈ J }. Corresponding to these functions we define NAp(·),q(·) (ε) to be the minimum of all those  n ∈ N such that I can be written as I = nj=1 Ij , where each Ij is a closed sub-interval of I , |Ii ∩ Ij | = 0 (i = j ) and Ap(·),q(·) (Ij )  ε for every j. The quantities NBp(·) (ε), NCp(·),q(·) (ε), NDp(·) (ε) are defined in an exactly similar way. We shall write Ap(·) (J ) = Ap(·),p(·) (J ) and Cp(·) (J ) = Cp(·),p(·) (J ), denoting these quantities by Ap , Cp respectively if p(x) = p is a constant function. When p(x) = p and q(x) = q are constant functions then we will write Ap,q (J ) = Ap(·),q(·) (J ) and Cp,q (J ) = Cp(·),q(·) (J ). Functions of this kind were introduced in previous work on the s-numbers of Hardy-type operators in the context of classical Lebesgue spaces (see, for example, [2,3,5] and [7]), and in fact for that situation we have the following result. Lemma 2.5. When J = (c, d) ⊂ I, and p is a constant function, so that p(x) = p ∈ (1, ∞) for all x ∈ I, 1/p |J |  , Ap (J ) = Bp (J ) = p  p p−1 2πp where πp =

2π p sin(π/p) .

The following lemma was proved in [13]. Lemma 2.6. Let J = (c, d) ⊂ I, and p, q ∈ (1, ∞). Then 1− p1 + q1

Tf q,J (p  + q) sup = f f p,J



and the extremals are the non-zero multiples of cosp,q (πp,q x/2). This leads us to the following result.

1− p1 + q1

(p  )1/q q 1/p |J | B(1/p  , 1/q)

,

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Lemma 2.7. Let J = (c, d) ⊂ I, and p, q ∈ (1, ∞). Then Ap,q (J ) = Cp,q (J ) =

1− p1 + q1

(p  + q)



1− p1 + q1

(p  )1/q q 1/p |J | 2B(1/p  , 1/q)

1− p1 + q1

:= B(p, q)|J |

(2.2)

From Lemma 2.5, by using techniques from [4,7,8] and with the help of the well-known inequality     bn (T )  min cn (T ), dn (T )  max cn (T ), dn (T )  an (T ),

(2.3)

we obtain the following theorem. Theorem 2.8. Let p be as in Lemma 2.5 and let T be viewed as a map from Lp (I ) to itself. Then for all n ∈ N, an (T ) = cn (T ) = dn (T ) = bn (T ) = where πp =

(p  p p−1 )1/p |I |, πp n

2π p sin(π/p) .

It is known that under the conditions of the last lemma, A(J ) depends continuously on the right-hand endpoint of J ; that is, with a slight abuse of notation, the function A(c, ·) is continuous. A similar result holds for non-constant p: this is formulated in the next lemma together with the corresponding results for B, C and D. Lemma 2.9. Let p(·), q(·) ∈ P(I ). Then the functions Ap(·),q(·) (c, t), Bp(·) (c, t), Cp(·),q(·) (c, t) and Dp(·) (c, t) of the variable t are non-decreasing and continuous. Analogously the functions Ap(·),q(·) (t, d), Bp(·) (t, d), Cp(·),q(·) (t, d) and Dp(·) (t, d) are non-decreasing and continuous. Proof. We start with A := Ap(·),q(·) . First we prove that A(c, d)  A(c, d + h) when h  0. Clearly   ·     A(c, d + h) = inf sup  f (t) dt    y∈(c,d+h)

; f p(·),(c,d+h)  1

q(·),(c,d+h)

y

  ·     = min inf sup  f (t) dt    y∈(c,d)

y

  ·     inf sup  f (t) dt    y∈(d,d+h) y

:= min{X, Y }.

; f p(·),(c,d+h)  1 , q(·),(c,d+h)

; f p(·),(c,d+h)  1 q(·),(c,d+h)

D.E. Edmunds et al. / Journal of Functional Analysis 257 (2009) 219–242

225

Now   ·     X  inf sup  f (t) dt    y∈(c,d) y

; f p(·),(c,d)  1 = A(c, d) q(·),(c,d)

and   ·     Y  inf sup  f (t) dt    y∈(d,d+h)

; f p(·),(c,d)  1

q(·),(c,d)

y

  ·      sup  f (t) dt   

; f p(·),(c,d)  1

q(·),(c,d)

d

  ·      inf sup  f (t) dt    y∈(c,d)

; f p(·),(c,d) = A(c, d),

q(·),(c,d)

y

which gives A(c, d + h)  A(c, d). Let us prove the continuity of A. By Hölder’s inequality (see [6]) we have, for some α  1 (independent of f , x and y), x       f (t) dt   α1p (·),(y,x) f p(·),(y,x)   y

and considering 1p (·),(y,x) as a function of x we obtain   1p (·),(y,x) 

q(·),(d,d+h)

 1p (·),(c,d+h) 1q(·),(d,d+h)

which gives A(c, d)  A(c, d + h)

  ·     = inf sup  f (t) dt    y∈(c,d+h) y

  ·      inf sup  f (t) dt    y∈(c,d+h) y

 ·      +  f (t) dt    y

; f p(·),(c,d+h)  1 q(·),(c,d+h)

q(·),(c,d)

; f p(·),(c,d+h)  1 q(·),(d,d+h)

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  ·      inf sup  f (t) dt    y∈(c,d+h)

q(·),(c,d)

y

  + α 1p (·),(y,x) f p(·),(y,x) 

q(·),(d,d+h)

; f p(·),(c,d+h)  1

  ·      inf sup  f (t) dt    y∈(c,d+h)

q(·),(c,d)

y

  + α 1p (·),(y,x) 

q(·),(d,d+h)

; f p(·),(c,d+h)  1

  ·      inf sup  f (t) dt    y∈(c,d+h) y



; f p(·),(c,d+h)  1

q(·),(c,d)

+ α1p (·),(c,d+h) 1q(·),(d,d+h)   ·

     inf sup  f (t) dt  ; f p(·),(c,d+h)  1   y∈(c,d) y

q(·),(c,d)

+ α1p (·),(c,d+h) 1q(·),(d,d+h)   ·

    = inf sup  f (t) dt  ; f p(·),(c,d)  1   y∈(c,d) y

q(·),(c,d)

+ α1p (·),(c,d+h) 1q(·),(d,d+h) = A(c, d) + α1p (·),(c,d+h) 1q(·),(d,d+h) . Since q(x) ∈ P(I ) we know that 1q(·),(d,d+h) → 0 as h → 0 and so, A(c, ·) is rightcontinuous. Left-continuity is proved in a corresponding manner, and the continuity of A(c, ·) follows. The arguments for B, C and D are similar. 2 As an immediate consequence of this and Lemma 2.3 we have Lemma 2.10. Let p(·) ∈ P(I ). Then T : Lp(·) (I ) → Lp(·) (I ) is compact and for all ε > 0 the quantities NAp(·) (ε), NBp(·) (ε), NCp(·) (ε) and NDp(·) (ε) are finite. We now have Lemma 2.11. Let p(·) be as in the last lemma. Then given any N ∈ N, there exists a unique ε > 0 such that NA (ε) = N, and there is a non-overlapping covering of I by intervals IAi (i = 1, . . . , N) such that A(IAi ) = ε for i = 1, . . . , N. The same result holds when A is replaced by B, C, D. Proof. The existence follows from the continuity properties established in Lemma 2.9.

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227

i N For uniqueness, observe that given two non-overlapping coverings of I , {IAi }N i=1 and {JA }i=1 , j there are m, j, k, l such that IAm ⊂ JA and JAk ⊂ IAl . Now, assuming A(IAi ) = ε1 , A(JAi ) = ε2 we obtain ε1  ε2  ε1 by the monotonicity of A. 2

3. The case when p(·) is a step-function Let {Ji }m i=1 be a disjoint covering of I by intervals and let p be the step-function defined by p(x) =

m 

(3.1)

χJi (x)pi ,

i=1

where each pi belongs to (1, ∞). For simplicity, in this section we shall write A instead of Ap(·) ; B, C, D will have the analogous meaning. Lemma 3.1. Let p(·) be the step-function given by (3.1). Then T : Lp(·) (I ) → Lp(·) (I ) is compact and for sufficiently small ε > 0, (i) bNC (ε)−m (T ) > ε, (ii) aNA (ε)+2m−1 (T ) < ε. Proof. Let ε > 0. The compactness of T follows from Lemma 2.10, as does the finiteness of NA (ε) and NC (ε). (i) By the continuity of C(c, ·), there exists a set of non-overlapping intervals {Ii : i = , . . . , NC (ε)} covering I and such that C(Ii ) = ε whenever 1  i < NC (ε) and C(INC (ε) )  ε. Let η ∈ (0, ε). Then corresponding to each i, 1  i < NC (ε), there is a function fi such that supp fi ⊂ Ii := (ai , ai+1 ), fi p(·) = 1, ε − η < Tfi p(·)  ε and (Tf )(ai ) = (Tf )(ai+1 ) = 0. By {Iik }M k=1 we denote the set of those intervals Ii , 1  i < NC (ε), each of which is contained in one of the intervals Jl from the definition (3.1) of p(·). Then NC (ε) − m  M  NC (ε). Define by XM = f =

M 

αir fir ; αir ∈ R

r=1

an M-dimensional subspace of Lp(·) (I ). Note that since p(·) is constant on Iir , p(x) = pir on Iir . Choose 0 = f ∈ XM . With λ0 := Tf p we have    M    Tf (x) p(x)   1  dx  λ0  I

r=1I

ir

   Tf (x) p(x)   dx  λ  0

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D.E. Edmunds et al. / Journal of Functional Analysis 257 (2009) 219–242

=

    M  M    pir pir 1 pir  ε − η pir  Tf (x) dx  f (x) dx λ0 λ0 r=1

=

  M     f (x) p(x)   dx =  λ /(ε − η)  0 r=1I

=

r=1

Iir

∪M r=1 Iir

ir

    f (x) p(x)   dx.  λ /(ε − η)  0

Iir

   f (x) p(x)   dx  λ /(ε − η)  0

I

Hence f p,I  Tf p,I /(ε − η), and so bNC (ε)−m (T )  bM (T )  ε − η. NA (ε) be a set of non(ii) This follows a pattern similar to that of (i). This time we let {Ii }i=1 overlapping intervals covering I for which A(Ii ) = ε for i = 1, . . . , NA (ε)−1 and A(INA (ε) )  ε. By {Ii+ }M i=1 we denote the family of all non-empty intervals for which there exist j and k such that Ii+ = Ij ∩ Jk . Clearly NA (ε)  M  NA (ε) + 2(m − 1). Let η > 0. Then given any i ∈ {1, 2, . . . , M} there exists yi ∈ Ii+ such that  ·      sup  f    yi

p,Ii+

: f p,I + = 1  ε + η. i

Define

Pε (f ) =

 yi M 

 f (x) dx χI + . i

i=1

a

Plainly Pε is a linear map from Lp(·) (I ) to Lp(·) (I ) with rank M. Let pi be the constant value of p(·) on Ii+ . Then we have for any λ0 ∈ (0, ∞) and f ∈ Lp(·) (I ),    M    (T − Pε )f (x) p(x)   dx =   λ0 i=1

I



Ii+

M 

  x p(x)  M   yi f  −pi   dx = λ0  λ  0

−pi

λ0

i=1

 (ε + η)pi

i=1

Now choose λ0 = (1 − η)(T − Pε )f p,I . Then

|f |pi dx = Ii+

x pi      f  dx  

Ii+ yi

    f (x) p(x)   dx.  λ /(ε + η)  0 I

D.E. Edmunds et al. / Journal of Functional Analysis 257 (2009) 219–242

     (T − Pε )f (x) p(x)   1<  dx   λ0 I

I

229

   f (x) p(x)   dx,  λ /(ε + η)  0

from which we see that   f p(·),I > (1 − η)(T − Pε )f p(·),I /(ε + η), so that ε + η (T − Pε )f p(·),I > . 1−η f p(·),I The proof is completed on letting η → 0.

2

Lemma 3.2. Let p(·) be the step-function given by (3.1). Then 1 lim εN(ε) = ε→0 2π



  1/p(x)   p (x)p(x)p(x)−1 sin π/p(x) dx,

I

where N stands for NA , NB , NC or ND . Proof. Simply use the fact that p(·) is a step-function together with Lemmas 2.5 and 2.11.

2

Finally we can give the main result of this section. Theorem 3.3. Let p(·) be the step-function given by (3.1). Then for the compact map T : Lp(·) (I ) → Lp(·) (I ) we have lim nsn (T ) =

n→∞

1 2π



  1/p(x)   p (x)p(x)p(x)−1 sin π/p(x) dx,

I

where sn denote the nth approximation, Gelfand, Kolmogorov or Bernstein number of T . Proof. Using Lemmas 3.1 together with inequalities (2.3), we have εNA (ε)  aNA (ε)+2m−1 (T )NA (ε)  bNA (ε)+2m−1 (T )NA (ε) and εNC (ε)  bNC (ε)−m (T )NC (ε). Now use Lemma 3.2 to obtain the result for the approximation and Bernstein numbers. The rest follows from (2.3) again. 2

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4. The case when p(·) is strongly log-Hölder-continuous To obtain a result in this case similar to that of Theorem 3.3 the idea is to approximate p by step-functions. This requires that control be kept of the changes in the various norms when p is replaced by an approximating function, and we begin by giving such a result, which has independent interest. Let p(·), q(·) ∈ P(I ) be such that for some ε ∈ (0, 1), 1 < p(x)  q(x)  p(x) + ε

for all x ∈ I.

(4.1)

We know from Theorem 2.1 that Lq(·) (I ) is continuously embedded in Lp(·) (I ); denote by id the norm of the corresponding embedding. Our object is to obtain upper and lower bounds for id in terms of ε. 4.1. Suppose that p(·) and q(·) satisfy (4.1) and that f ∈ M(I ) is such that Lemma q(x) |f (x)| dx  1 Then I 

  f (x)p(x) dx  ε|I | + ε −ε .

I

Proof. Set I1 = {x ∈ I : |f (x)| < ε}, I2 = {x ∈ I : ε  |f (x)|  1} and I3 = {x ∈ I : 1 < |f (x)|}. Then 

3    f (x)p(x) dx =



j =1 I

I

3    f (x)p(x) dx = Aj ,

say.

j =1

j

Evidently  A1 

 ε p(x) dx 

I1

ε dx  ε|I |

(4.2)

I1

and  A3 

  f (x)q(x) dx.

I3

Since ε  |f (x)|  1 on I2 and ε < 1 we have, by (4.1), q(x)−p(x)  ε ε  ε q(x)−p(x)  f (x) 1 on I2 , and so p(x)−q(x)   ε −ε . 1  f (x) Hence

(4.3)

D.E. Edmunds et al. / Journal of Functional Analysis 257 (2009) 219–242

 A2 =

    f (x)q(x) f (x)p(x)−q(x) dx  ε −ε

I2



231

  f (x)q(x) dx.

(4.4)

I2

Now (4.2), (4.3) and (4.4) give 

  f (x)p(x) dx  ε|I | + ε −ε

I



  f (x)q(x) dx +

I2

 ε|I | + ε

−ε

 ε|I | + ε −ε

 

 I3

  f (x)q(x) dx +

I2

  f (x)q(x) dx 

  f (x)q(x) dx



I3

  f (x)q(x) dx  ε|I | + ε −ε ,

I

2

as required.

Lemma 4.2. Suppose that p(·) and q(·) satisfy (4.1). Then id  ε|I | + ε −ε .  Proof. Observe that K := ε|I | + ε −ε > 1. Given any f such that I |f (x)|q(x) dx  1, by Lemma 4.1 we see that          f (x)/K p(x) dx  f (x)/K 1/p(x) p(x) dx = K −1 f (x)p(x) dx I

Thus id  K.

I

I

   ε|I | + ε −ε /K = 1. 2

We now turn to lower bounds. Lemma 4.3. Suppose that p(·) and q(·) satisfy (4.1) and that |I |  1. Then id  1. Proof. Define a function g by g(x) = |I |−1/q(x) (x ∈ I ). Then |I |−p(x)/q(x)  |I |−1 , we have for each λ ∈ (0, 1),  I

  g(x)/λp(x) dx =



|I |−p(x)/q(x) dx  λp(x)

I



|I |−1 dx  λp(x)

I

Hence id  λ for each λ ∈ (0, 1), and so id  1.



 I

|g(x)|q(x) dx = 1. Since

|I |−1 1 dx = > 1. λ λ

I

2

Lemma 4.4. Suppose that p(·) and q(·) satisfy (4.1) and that |I | < 1. Then id  |I |ε . Proof. Again we consider the function g(x) = |I |−1/q(x) : 1− p(x) q(x)

|I |

= |I |

q(x)−p(x) q(x)

 I

|g(x)|q(x) dx = 1. Since

 |I |ε/q(x)  |I |ε ,

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we have 

  g(x)p(x) dx =

I



− p(x) q(x)

|I |

dx = |I |−1

I



1− p(x) q(x)

|I |

dx  |I |ε .

I

Thus, for each positive λ < |I |ε ,  I

  g(x)/λp(x) dx >

     g(x) p(x)   dx   |I |ε  I

= |I |−ε



I

   g(x) p(x)   dx  |I |ε/p(x) 

  g(x)p(x) dx  |I |−ε |I |ε = 1.

I

It follows that id  λ for each λ < |I |ε , which gives the result.

2

Putting these results together we have the following theorem and corollary. Theorem 4.5. Suppose that p(·) and q(·) satisfy (4.1). Then   min 1, |I |ε  id  ε|I | + ε −ε . Corollary 4.6. Let p(·) ∈ P(I ) and suppose that for each n ∈ N, qn (·) ∈ P(I ) and εn > 0, where limn→∞ εn = 0, and for all n ∈ N and all x ∈ I, 1 < p(x)  qn (x)  p(x) + εn . Denote by idn the natural embedding of Lqn (·) (I ) in Lp(·) (I ). Then lim idn  = 1.

n→∞

In the next, we prove a few technical lemmas. Lemma 4.7. Let δ > 0 and let J ⊂ I be an interval and p(·), q(·) ∈ P(J ). Assume p(x)  q(x)  p(x) + δ in J . Then  −2 2  δ|J | + δ −δ Ap(·)+δ,p(·) (J )  Aq(·) (J )  δ|J | + δ −δ Ap(·),p(·)+δ (J ). Proof. Set   B1 = f ; f q(·)  1 ,

  B2 = f ; f p(·)  δ|J | + δ −δ ,

where the norms are with respect to the interval J . By Theorem 4.5 we have f p(·)  (δ|J | + δ −δ )f q(·) which gives B1 ⊂ B2 and

D.E. Edmunds et al. / Journal of Functional Analysis 257 (2009) 219–242

 ·      Aq(·) (J ) = inf sup  f    y∈J y

 ·      ; f q(·)  1 = inf sup  f    y∈J

233

q(·)

y

 ·      −δ   inf sup δ|J | + δ  f    y∈J

y

; f ∈ B1 q(·)

; f ∈ B2 p(·)+δ

  ·     f   −δ 2 = δ|J | + δ inf sup    δ|J | + δ −δ  y∈J

p(·)+δ

y

 ·        −δ 2 = δ|J | + δ inf sup  g    y∈J y

    f   ; −δ δ|J | + δ 

1

p(·)

; gp(·)  1

p(·)+δ

2  = δ|J | + δ −δ Ap(·),p(·)+δ (J ) The second part of the inequality can be proved analogously.

2

Lemma 4.8. Let an interval J ⊂ I, with |J |  1, and p ∈ (1, ∞) be given. Then there exists a bounded positive function η defined on (0, 1), with η(δ) → 0 as δ → 0, such that if p(·), q(·) ∈ P(J ) with p  p(x)  p + δ,

p  q(x)  p + δ

in J,

then    Ap(·) (J )   1 + η(δ) |J |−2δ . 1 − η(δ) |J |2δ  Aq(·) (J ) Proof. It suffices to prove only the right-hand part of the inequality. By Lemma 4.7 and (2.2) we have 1

1

4 Ap,p+δ (J )  4 B(p, p + δ) |J |1− p + p+δ Ap(·) (J )   δ|J | + δ −δ = δ|J | + δ −δ 1 + p1 Aq(·) (J ) Ap+δ,p (J ) B(p + δ, p) |J |1− p+δ −2δ 4 B(p, p + δ) 4 B(p, p + δ) −2δ   |J | p(p+δ)  δ|J | + δ −δ |J | . = δ|J | + δ −δ B(p + δ, p) B(p + δ, p)

Since  4 B(p, p + δ) lim δ|J | + δ −δ = 1, δ→0 B(p + δ, p) we can choose η(δ) such that

 4 B(p, p + δ) η(δ) := max δ, δ|J | + δ −δ −1 B(p + δ, p) to establish our assertion.

2

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Lemma 4.9. Let δ > 0, a1 < b1  a2 < b2 and Ji = (ai , bi ) ⊂ I , i = 1, 2. Assume that f1 , f2 are functions on I such that supp fi ⊂ Ji , i = 1, 2, and Tf1 p(·),J1 > δ. Then   T (f1 − f2 )

p(·),I

> δ.

Proof. Since T (f1 /δ)p(·),J1 > 1 we have p(x) b1 x   f1 (t)  dt  dx > 1.    δ

a1

a1

Then p(x)  b  b x   T (f1 − f2 )(x) p(x) f (t) − f (t) 1 2     dt  dx =  dx     δ δ a

a

a

a1 =

b1 ... +

a

a2 ... +

a1

b2 ... +

b1

b ... +

a2

... b2

p(x) b1x   f1 (t) − f2 (t)  dt  dx     δ a1

a

p(x) b1x   f1 (t)  =  dt  dx > 1   δ a1

and so T (f1 − f2 )p(·),I > δ.

a

2

Next we recall the well-known concept of a log-Hölder continuous function which is widely used in the theory of variable exponent spaces. Following current terminology we shall say that p(·) is log-Hölder continuous if there is a positive constant L such that −|p(x) − p(y)| ln |x − y|  L for all x, y ∈ I with 0 < |x − y| < 12 . In what follows we will require a little stronger condition on the function p(·) defined on I . We remind the reader that I = [a, b] is a compact interval. Definition 4.10. Let p(·) ∈ P(I ). We say that p(·) is strong log-Hölder continuous (and write p(·) ∈ SLH(I )) if there is an increasing continuous function ψ(t) defined on [0, |I |] such that lim ψ(t) = 0 and t→0+

    −p(x) − p(y) ln |x − y|  ψ |x − y|

for all x, y in I with 0 < |x − y| < 1/2.

It is easy to see any Lipschitz or Hölder function p(·) is SLH(I ).

(4.5)

D.E. Edmunds et al. / Journal of Functional Analysis 257 (2009) 219–242

235

Proposition 4.11. Let p(·) ∈ P(I ) be strong log-Hölder continuous on I. Then 1 lim εN(ε) = ε→0 2π



  1/p(x)   p (x)p(x)p(x)−1 sin π/p(x) dx,

I

where N stands for NAp(·) or NCp(·) . Proof. We prove only the case N = NAp(·) , the case N = NCp(·) follows by a simple modification. Let N ∈ N. By Lemma 2.11 there exists a constant εN > 0 and a set of non-overlapping intervals N {IiN }N i=1 covering I such that Ap(·) (Ii ) = εN for every i. Define a step-function qN (x) by qN (x) =

N 

p +N χI N (x).

i=1

Ii

i

and set δN,i = p +N − p −N . Ii

Ii

Then p(x)  qN (x)  p(x) + δN,i

for all i = 1, 2, . . . , N.

Claim 1. εN → 0 as N → ∞. Proof. Clearly, εN is non-increasing. Assume for a moment that there exists δ > 0 such that εN > δ for all N . Fix N and denote IiN := Ii = (ai , ai+1 ). Since Ap(·),Ii > δ there are fi , with · supp fi ⊂ Ii , such that fi p(·),Ii  1 and  ai fi p(·),Ii = Tfi p(·),Ii > δ for i = 1, . . . , N . By Lemma 4.9,   T (fi − fj ) > δ for i < j p(·),I and so, we have found N functions f1 , f2 , . . . , fN from the unit ball such that   T (fi − fj ) > δ. p(·),I The fact that N can be arbitrary contradicts the compactness of T .

2

Claim 2. limN →∞ max{|IiN |; i = 1, 2, . . . , N} = 0. Proof. Assume the contrary. Then there are sequences Nk , ik ∈ {1, 2, . . . , Nk } and an interval J such that J ⊂ IiNk k and so,  N  εNk = Ap(·) Iik k  Ap(·) (J ) > 0 which contradicts the fact that εN → 0.

2

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Claim 3. There is a sequence βN 1 such that  2δ  −2δ   −1 εN IiN  N,i  AqN (·) IiN  βN εN IiN  N,i βN holds for all i ∈ {1, 2, . . . , N}. Proof. Since p −  qN (x), p(x)  p − + δN,i on IiN we have by Lemma 4.8,    2δ  −2δ Ap(·) (IiN ) 1 − η(δN,i ) IiN  N,i   1 + η(δN,i ) IiN  N,i . N AqN (·) (Ii ) Now, using εN = Ap(·) (IiN ) we have  N 2δN,i  N −2δN,i   εN εN I  I   AqN (·) IiN  i 1 + η(δN,i ) 1 − η(δN,i ) i and the assertion follows.

2

Claim 4. The inequality  N −δN,i N I   eψ(|Ii |) i holds for all N and i ∈ {1, 2, . . . , N}. Proof. Fix IiN . Since p(·) ∈ SLH(I ) we know that p(·) is a continuous function on I . Because p +N − p −N = δN,i there are points x, y ∈ IiN with |p(x) − p(y)| = δN,i . Using (4.5) we obtain Ii

Ii

 N −δN,i N I   |x − y|−|p(x)−p(y)|  eψ(|x−y|)  eψ(|Ii |) . i

2

Claim 5. There is a constant C > 0 such that the inequality   C −1 εN  IiN   CεN holds for all N and i ∈ {1, 2, . . . , N}. Proof. Remark that qN (·) = p −N + δN,i := rN,i is a constant function on IiN and so, by Lemma 2.7,

Ii

    AqN (·) IiN = B(rN,i , rN,i )IiN . It is easy to see that there is a > 0 such that a −1  B(rN,i , rN,i )  a holds for all N and i ∈ {1, 2, . . . , N}. Using Claim 4 we have  N −2δN,i N I   e2ψ(|Ii |)  e2ψ(|I |) := K, i

D.E. Edmunds et al. / Journal of Functional Analysis 257 (2009) 219–242

237

and by Claim 3,   −1 K −1 βN εN  B(rN,i , rN,i )IiN   KβN εN . Hence   −1 a −1 K −1 βN εN  IiN   aKβN εN . Since βN 1, the assertion follows.

2

Now, since by Claim 1, εN → 0 we know by Claim 5 that max{|IiN |; i = 1, 2, . . . , N } → 0 as N → ∞ and by Claim 4,  N −2δN,i N N I   e2ψ(|Ii |)  e2ψ(max{|Ii |; i=1,2,...,N }) := γN 1. i Setting αN = βN γN we obtain αN 1, and by Claim 2 we have   −1 αN εN  AqN (·) IiN  αN εN . Moreover, by Claim 5 we have N εN = C

N  i=1

C −1 εN  C

N   N I  = C|I | i

i=1

which gives, by (4.6), N N  −1  −1       NεN αN αN εN − εN  −1 = AqN (·) IiN − N εN i=1



i=1

N 

(αN εN − εN ) = N εN (αN − 1).

i=1

Consequently, N     N   AqN (·) Ii − N εN  → 0 as N → ∞.    i=1

On the other hand we have by Lemma 2.5 (recall again that qN (·) is constant on IiN ), N 

N   1/    1   qN (·)qN (·)qN (·)−1 qN (·) sin π/qN (·) IiN  AqN (·) IiN = 2π i=1 i=1    1/p(x)   1 → p (x)p(x)p(x)−1 sin π/p(x) dx, 2π I

(4.6)

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D.E. Edmunds et al. / Journal of Functional Analysis 257 (2009) 219–242

and so, 

1 lim N εN = N →∞ 2π

  1/p(x)   p (x)p(x)p(x)−1 sin π/p(x) dx.

I

Since εN is monotone it is not difficult to see limN →∞ N εN = limε→0 εN (ε) and consequently 1 lim εN(ε) = ε→0 2π





p  (x)p(x)p(x)−1

1/p(x)

  sin π/p(x) dx.

2

I

Given p(·) ∈ SLH(I ), we construct step-functions that are approximations to p(·). Let N ∈ N and use Lemma 2.11, applied to the function D := Dp(·) : there exists ε > 0 such that ND (ε) = N and there are non-overlapping intervals IiD (i = 1, . . . , N ) that cover I and are such that D(IiD ) = ε for i = 1, . . . , N. Define + pD,N (x) =

N  i=1

pI+D χI D (x), i

− pD,N (x) =

i

N 

pI−D χI D (x);

i=1

i

i

+ − step-functions pB,N (·) and pB,N (·) are defined in an exactly similar way, with the function B in place of D and with intervals IiB arising from the use of that part of Lemma 2.11 related to B.

Lemma 4.12. Let p(·) ∈ P(I ) and N ∈ N. Let ε > 0 correspond to N in the sense of Lemma 2.11, applied to B, so that NB (ε) = N , and write − p − (x) = pB,N (x),

+ p + (x) = pB,N (x),

− + where pB,N (·) and pB,N (·) are defined as indicated above. Then

  − + aN +1 T : Lp (·) (I ) → Lp (·) (I )  ε. Proof. In the notation of Lemma 2.11, there are intervals IiB such that B(IiB ) = ε for i = 1, . . . , N. For each i there exists yi ∈ IiB such that  ·     B   B Ii = sup  f    yi

: f p− ,I B  1 . i

p + ,IiB

Define N  

yi

PN f (x) =

f (y) dy · χI B (x); i

i=1 a

D.E. Edmunds et al. / Journal of Functional Analysis 257 (2009) 219–242

plainly PN has rank N. Let f ∈ Lp

− (·)

239

(I ) and set λ0 = εf p− ,I .

(4.7)

Then  −   N    f (x) p (x)   1=  dx = λ0 /ε  i=1

I

 −   f (x) p (x)   dx.  λ /ε  0

IiB

Recall that on IiB the functions p − (·) and p + (·) have constant values pi− , pi+ , say, respectively, with pi+ /pi−  1. Thus  − p+ /p−  p+ /p− N   N   i i i i p −  f (x) pi  pi+ i     f (x) = (ε/λ0 ) dx . 1  λ /ε  dx 0 i=1

i=1

IiB

IiB

Use of the fact that p+ 1/p+     x 1/p− i   i i p −    i   f (y) ε = sup / dy  f (y) dy  dx   f IiB yi

IiB

now gives  N  + (1/λ0 )pi 1 i=1

x p + N    i    f (y) dy dx =     i=1

IiB yi

 x  +  yi f (y) dy pi   dx   λ0

IiB

 +    (T − PN )(f )(x) p (x)   dx, =   λ0 I

from which it follows that (T − PN )f p+ ,I  λ0 . Using the definition (4.7) of λ0 we see that   (T − PN )f  +  εf p− ,I , p ,I and so aN +1 (T : Lp

− (·)

(I ) → Lp

+ (·)

(I ))  ε, as claimed.

2

We next obtain a lower estimate for the Bernstein numbers. Lemma 4.13. Let p(·) ∈ P(I ) and N ∈ N. Let ε > 0 correspond to N in the sense of Lemma 2.11, applied to D, so that ND (ε) = N, and write − (x), p − (x) = pD,N

+ p + (x) = pD,N (x),

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D.E. Edmunds et al. / Journal of Functional Analysis 257 (2009) 219–242

− + where pD,N (·) and pD,N (·) are defined as indicated above. Then

  + − bN T : Lp (·) (I ) → Lp (·) (I )  ε. Proof. In the notation of Lemma 2.11, there are intervals IiD such that D(IiD ) = ε for i = + 1, . . . , N. Since T is compact, for each i there exists fi ∈ Lp (IiD ) with supp fi ⊂ IiD , Tfi p− ,I D /fi p+ ,I D = ε, i

(4.8)

i

and Tfi (ci ) = Tfi (ci+1 ) = 0, where ci and ci+1 are the endpoints of IiD . On each IiD the functions p − (·) and p + (·) are constant; denote these constant values by pi− and pi+ , respectively and note that pi− /pi+  1. Set XN = f =

N 

αi fi ; αi ∈ R .

i=1

Then dim XN = N . Choose any non-zero f ∈ XN and set λ0 = εf p+ ,I . Then 1=

 +    f (x) p (x)   dx  λ /ε  0 I

 + N     f (x) p (x)   = dx  λ /ε  i=1



IiD

0

 + p− /p+ N    i i  f (x) pi    λ /ε  dx 0 i=1

IiD

 p N  i p +  pi− i   f (x) = (ε/λ0 ) dx

−

i=1

=

/pi+

IiD

 N   +  − αi fi (x)pi dx)pi− /pi+ . (ε/λ0 )pi i=1

IiD

Use of (4.8) now shows that pi−   N N     −     − Tf (x) p pi i     T (αi fi )(x) 1 (1/λ0 ) dx =  λ  dx = i=1

IiD

from which it follows that

i=1

IiD

0

I

 p −  Tf (x)  (x)   dx  λ  0

D.E. Edmunds et al. / Journal of Functional Analysis 257 (2009) 219–242

241

  + − ε  bN T : Lp (·) (I ) → Lp (·) (I ) , and the proof is complete.

2

Theorem 4.14. Let p(·) ∈ P(I ) be continuous on I . For all N ∈ N denote by εN numbers satisfying N = NB (εN ). Then there are sequences KN , LN with KN → 1, LN → 1 as N → ∞ such that (i) aN +1 (T : Lp(·) (I ) → Lp(·) (I ))  KN εN , (ii) bN (T : Lp(·) (I ) → Lp(·) (I ))  LN εN . Proof. First we prove (i). In view of the property (2.1) of the approximation numbers, aN +1 (T : Lp(·) (I ) → Lp(·) (I )) is majorized by  − p(·)  − id : L (I ) → LpB,N (I ) N     − + + pB,N × aN +1 T : LpB,N (I ) → LpB,N (I ) × id+ (I ) → Lp(·) (I ), N :L where id− and id+ are the obvious embedding maps. When N → ∞, since |IiB | → 0 and p(·) is continuous, it is clear that     p(·) − p − (·) → 0 and p(·) − p + (·) → 0. B,N B,N ∞ ∞ + − (·), pB,N (·) are the same as in Lemma 4.12.) Thus by Corollary 4.6, (Here IiB , pB,N

 − p(·)  − id : L (I ) → LpB,N (I ) → 1, N

 + p+  id : L B,N (I ) → Lp(·) (I ) → 1 N

as N → ∞. The result now follows from Lemma 4.12. To prove (ii) we follow the idea of the proof of (i) with the help of Lemma 4.13.

2

Theorem 4.15. Let p ∈ SLH(I ). Then 1 lim nsn (T ) = n→∞ 2π



  1/p(t) p (t)p(t)p(t)−1 sin



 π dt, p(t)

I

where sn denote the nth approximation, Gelfand, Kolmogorov or Bernstein number of T . Proof. Use Theorem 4.14, Proposition 4.11 and the inequality (2.3).

2

It is not difficult to combine proofs of Theorem 3.3 and Theorem 4.15 to obtain the following theorem, which contains both these results. Theorem 4.16. Let Ji , i = 1, 2, . . . , m, be a finite decomposition of I . Assume that p(·) be such that p(·) ∈ SLH(Ii ) for each i ∈ {1, 2, . . . , m}. Then

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lim nsn (T ) =

n→∞

1 2π





p  (t)p(t)p(t)−1

1/p(t)

 sin

 π dt, p(t)

I

where sn denote the nth approximation, Gelfand, Kolmogorov or Bernstein number of T . References [1] D.E. Edmunds, W.D. Evans, Spectral Theory and Differential Operators, Oxford University Press, 1987. [2] D.E. Edmunds, W.D. Evans, Hardy Operators, Function Spaces and Embeddings, Springer-Verlag, Berlin, 2004. [3] D.E. Edmunds, J. Lang, Approximation numbers and Kolmogorov widths of Hardy-type operators in a nonhomogeneous case, Math. Nachr. 279 (2006) 727–742. [4] D.E. Edmunds, J. Lang, Behaviour of the approximation numbers of a Sobolev embedding in the one-dimensional case, J. Funct. Anal. 206 (2004) 149–166. [5] D.E. Edmunds, J. Lang, Bernstein widths of Hardy-type operators in a non-homogeneous case, J. Math. Anal. Appl. 325 (2007) 1060–1076. [6] O. Ková˘cik, J. Rákosník, On spaces Lp(x) (Ω) and W k,p(x) (Ω), Czechoslovak Math. J. 41 (1991) 592–618. [7] J. Lang, Improved estimates for the approximation numbers of Hardy-type operators, J. Approx. Theory 121 (2006) 61–70. [8] J. Lang, Estimates for n-widths of the Hardy-type operators, J. Approx. Theory 140 (2) (2006) 141–146. [9] G.G. Lorentz, V. Golitschek, Yu. Makovoz, Constructive Approximation: Advanced Problems, Springer-Verlag, Berlin, 1996. [10] A. Pietsch, Bad properties of the Bernstein numbers, Studia Math. 184 (2008) 263–269. [11] A. Pinkus, n-Widths in Approximation Theory, Springer-Verlag, Berlin, 1985. [12] M. R˚užiˇcka, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math., vol. 1748, Springer-Verlag, Berlin, 2000. [13] E. Schmidt, Über die Ungleichung, welche die Integrale über eine Potenz einer Funktion und über eine andere Potenz ihrer Ableitung verbindet, Math. Ann. 117 (1940) 301–326 (in German).

Journal of Functional Analysis 257 (2009) 243–270 www.elsevier.com/locate/jfa

Computing the first eigenvalue of the p-Laplacian via the inverse power method Rodney Josué Biezuner ∗ , Grey Ercole, Eder Marinho Martins Departamento de Matemática – ICEx, Universidade Federal de Minas Gerais, Av. Antônio Carlos 6627, Caixa Postal 702, 30161-970, Belo Horizonte, MG, Brazil Received 28 October 2008; accepted 13 January 2009 Available online 12 February 2009 Communicated by H. Brezis

Abstract In this paper, we discuss a new method for computing the first Dirichlet eigenvalue of the p-Laplacian inspired by the inverse power method in finite dimensional linear algebra. The iterative technique is independent of the particular method used in solving the p-Laplacian equation and therefore can be made as efficient as the latter. The method is validated theoretically for any ball in Rn if p > 1 and for any bounded domain in the particular case p = 2. For p > 2 the method is validated numerically for the square. © 2009 Elsevier Inc. All rights reserved. Keywords: p-Laplacian; First eigenvalue; Comparison principle; Power method

1. Introduction In finite dimensional linear algebra, the power method is often used in order to compute the first eigenvalue of invertible linear operators defined on Euclidean spaces (see [9], for instance). Briefly, given an invertible linear operator L, one picks a vector x and forms the sequence x, Ax, A2 x, . . . . In order to produce this sequence, it is not necessary to get the powers of A explicitly, since each vector in the sequence can be obtained from the previous one by multiplying it by A. It is * Corresponding author.

E-mail addresses: [email protected] (R.J. Biezuner), [email protected] (G. Ercole), [email protected] (E.M. Martins). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.01.023

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easy to show that the sequence converges, up to scaling, to the dominant eigenvector. Thus the largest eigenvalue can be found. In order to obtain the first (and therefore smallest) eigenvalue, one considers instead powers of A−1 , a method which is sometimes called the inverse power method or inverse iteration. More specifically, if {yn }n∈N denotes the normalized sequence of vectors produced by the inverse power method, then the first eigenvalue λ1 of A can be explicitly given as (see [5])   λ1 = lim yn · A−1 yn , n→∞

where · denotes the Euclidean inner product. In this work we carry this idea further, to nonlinear operators in infinite dimensional spaces, in order to develop a numerical method to compute the first eigenvalue of the nonlinear degenerate elliptic p-Laplacian operator. In order to describe the technique, first we establish some notation and recall some well-known results. Throughout this paper, Ω will denote a smooth bounded region in RN , N  1, and p will denote the p-Laplacian operator, that is,   p u = div |∇u|p−2 ∇u for 1 < p < ∞. We consider the Dirichlet eigenvalue problem for the p-Laplacian  −p u = λ|u|p−2 u in Ω, u=0 on ∂Ω.

(1)

(2)

As it is well known, the first eigenvalue λp (Ω) is positive, simple, can be variationally characterized, and the corresponding eigenfunction belongs to the Hölder space C 1,α (Ω), does not change sign and therefore can be taken positive. If p = 2, when p becomes the Laplacian operator , the value of λp (Ω) is well known for domains with simple geometry; for more general domains it can be determined by several methods. For p > 1 and N = 1, the value of λp (Ω) is known: if Ω = (a, b), then   πp p λp (Ω) = (p − 1) , b−a where 1 πp := 2 0

ds π/p . =2 √ p sen(π/p) 1 − sp

However, if p = 2 and N  2, the value of λp (Ω) is not explicitly known, not even for simple domains such as a ball or a square. In such cases, there are few available numerical methods for finding λp (Ω). In the absence of an exact value or even a good approximation for λp (Ω), lower bounds play an important role in its estimation, being of special interest in the literature (upper bounds are more easily obtainable from the variational characterization of λp (Ω)). An important lower bound for λp (Ω) (see [7]) is λp (B) where B ⊂ RN is a ball centered at the origin and with the same N -dimensional Lebesgue measure of Ω. We propose a method based on the inverse power method for obtaining λp (Ω) and prove its applicability in the case when Ω = B (without loss of generality we choose B to be the unit ball).

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We believe this specific result is very relevant and certainly will contribute to research on quasilinear problems in which spherical geometry or the estimation of λp (Ω) are important. Moreover, the main results that support the method are valid for a general domain (even unbounded) and thus they lead us to conjecture that our method is applicable to a more general class of domains. In the remainder of this introduction we describe in more detail the main results of this paper and our conjecture. 1,p Let W0 (Ω) denote the standard Sobolev space with norm u W 1,p (Ω) = ∇u Lp (Ω) . We 0 1,p recall that u ∈ W0 (Ω) is a weak solution to the homogeneous Dirichlet problem 

−p u = f u=0

in Ω, on ∂Ω,

(3)



for a given function f ∈ Lp (Ω), where p  = p/(p − 1) denotes the conjugate exponent of p, if 

 |∇u|p−2 ∇u · ∇v dx =

Ω

f v dx

(4)

Ω

1,p

for every test function v ∈ W0 (Ω). It is well known that if f ∈ C 0 (Ω) then the corresponding 1,p weak solution u belongs to C 1,α (Ω) and the inverse operator (−p )−1 : C 0 (Ω) → W0 (Ω) ∩ C 1,α (Ω) → C 0 (Ω) is continuous and compact. The technique runs as follows. First, iteratively define a sequence of functions (φn )n∈N ⊂ 1,p W0 (Ω) ∩ C 1.α (Ω) by setting φ0 ≡ 1 and, for n = 1, 2, 3, . . . , letting φn be the solution to the Dirichlet problem 

p−1

−p φn = φn−1 φn = 0

in Ω, on ∂Ω.

(5)

Then, for n  1, define the following sequences of real numbers   φn p−1 γn := inf Ω φn+1

(6)

and Γn := sup Ω

   φn  φn  = . φn+1  φn+1 L∞ (Ω)

(7)

We show that sequence (6) is indeed well defined and bounded above by the first Dirichlet eigenvalue of the p-Laplacian λp . This result in itself is of particular importance, since lower bounds for λp are hard to obtain. For sequence (7), we give strong evidence that it is well defined for at least all sufficiently large n (we prove this is true for balls and for general domains in the special case p = 2) and that it is bounded below by λp . We also show that sequence (6) is increasing (which implies among other things that we can obtain successively better lower bounds for λp ),

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whereas sequence (7) is decreasing. Thus, the limit γ := lim γn exists and is finite, the limit Γ := lim Γn exists and is likely finite (it is definitely finite in some cases), and they satisfy γ  λp  Γ. We conjecture that λp = γ = Γ.

(8)

If this conjecture is true, we also show that the first eigenfunction might be construed as the limit of a certain scaling of sequence (φn ). We are able to show that this indeed happens in the special cases of balls, for general p, and general domains, for p = 2. In the latter case we use the Hilbert space structure of the space W01,2 (Ω) and some well-known results for generalized Fourier expansions in eigenfunctions of the Laplacian. Needless to say, such an argument does not work for p = 2. However, numerical experiments indicate that for general p we do have γ =Γ and that these values approach known values of λp obtained by other authors using different techniques. We also consider the sequence of numbers  νn =

φn Lp (Ω)

φn+1 Lp (Ω)

p−1 (9)

and show that it is bounded below by λp and above by the sequence (Γn ), so that it is also our (independent) conjecture that λp = ν := lim νn .

(10)

In numerical experiments, it is observed that the convergence of this sequence is significantly faster than the convergences of the above two sequences. This paper is organized as follows. In Section 2, we prove the monotonicity of sequences (γn ) and (Γn ), and that the first eigenvalue is an upper bound for sequence (γn ). We also study sequence (νn ) and find that the first eigenvalue is a lower bound for this sequence as well as for sequence (Γn ). In Section 3, we show that if conjecture (8) is true, a certain scaled limit of sequence (φn ) might approach the first eigenfunction. Section 4 gives a complete proof of conjecture (8) in the case of n-dimensional balls. In Section 5, we prove the convergence of all three sequences to the first eigenvalue in the case p = 2. Section 6 shows that in principle this technique can be extended to yield higher eigenvalues and corresponding eigenfunctions, at least in the p = 2 case. Finally, in Section 7 we describe the numerical experiments performed and their results.

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2. Behavior of sequences (γn ), (Γn ) and (νn ) In the following, we denote by u ∞ and u p respectively the L∞ and Lp norms of a 1,p function u in Ω. Let (φn )n∈N ⊂ W0 (Ω) ∩ C 1.α (Ω) be the sequence defined by φ0 ≡ 1 and, for n  1, φn is the solution to the Dirichlet problem 

p−1

−p φn = φn−1 φn = 0

in Ω, on ∂Ω.

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2.1. Sequence (γn ) Set   φn p−1 . γn := inf Ω φn+1 First we show that sequence (γn ) is well defined. In the sequel we will resort several times to the well-known comparison principle for the p-Laplacian (see [4], for instance). For easiness of reference we state it here. Proposition 2.1 (Comparison principle). Let u1 , u2 ∈ C 1.α (Ω) satisfy 

−p u1  −p u2 u1  u2

in Ω, on ∂Ω.

Then, u1  u2 in Ω. Proposition 2.2. The sequence (φn )n∈N satisfies 0 < φn  φ1 ∞ φn−1

in Ω

for every n  1. Proof. First we show that φn is positive by using a simple argument involving the comparison principle. The comparison principle already implies that φn  0. Fix an arbitrary x0 ∈ Ω and let R > 0 be such that the ball BR (x0 ) centered at x0 with radius R is contained in Ω. We will show that φn (x) > 0 for all x ∈ BR (x0 ). Let ψ1 be the solution to the Dirichlet problem 

−p ψ1 = 1 in BR (x0 ), on ∂BR (x0 ). ψ1 = 0

It is well known that the solution to this problem is radially symmetric, that is, ψ1 (x) = v1 (r) for r = |x − x0 |, where v1 satisfies the problem 

p−2   v1 = r N −1 − r N −1 v1

 v1 (0) = v1 (R) = 0.

for 0 < r < R,

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Integrating the differential equation we find 1 R r  N −1 p−1 s ds dr v1 (r) = r

r

0

and hence v1 > 0 for 0 < r < R. Thus, since 

−p φ1 = −p ψ1 φ1  0 = ψ1

in BR (x0 ), on ∂BR (x0 ),

it follows from the comparison principle that φ1  ψ1 > 0 in BR (x0 ). For n  2 define the sequence (ψn ) iteratively by 

p−1

−p ψn+1 = ψn ψn+1 = 0

in BR (x0 ), on ∂BR (x0 ).

As before, ψn+1 (x) = vn+1 (r) where vn+1 is the solution to the (nonlinear) problem 

 p−2   p−1

− r N −1 vn+1 vn+1 = r N −1 vn  vn+1 (0) = vn+1 (R) = 0,

for 0 < r < R,

so that 1 R  r  N −1 p−1 s p−1 ψn+1 (x) = vn (s) ds dr > 0 for x ∈ BR (x0 ). r |x−x0 |

0

Assuming by induction that φn  ψn in BR (x0 ), it follows that 

p−1

p−1

−p φn+1 = φn  ψn φn+1  0 = ψn+1

= −p ψn+1

in BR (x0 ), on ∂BR (x0 ),

which implies that φn+1  ψn+1 in BR (x0 ). Therefore, we conclude that φn+1  ψn+1 > 0 in BR (x0 ) for all n. Since x0 is arbitrary, it follows that φn > 0 in Ω for all n. It remains to show that φn  φ1 ∞ φn−1 . Again we use an induction argument together with the comparison principle. Trivially, φ1  φ1 ∞ = φ1 ∞ φ0 . Assume φn  φ1 φn−1 .

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We have 

  p−1 p−1 p−1 −p φn+1 = φn  φ1 ∞ φn−1 = −p φ1 ∞ φn in Ω, φn+1 = 0 = φ1 ∞ φn on ∂Ω,

whence φn+1  φ1 ∞ φn .

2

As a consequence of Proposition 2.2, it follows that

φn+1 ∞  φ1 ∞ φn ∞ and φn 1  > 0. φn+1 φ1 ∞ Therefore, the sequence   φn p−1 γn = inf Ω φn+1

(12)

 p−1 1 1 γ0 = inf = . p−1 Ω φ1

φ1 ∞

(13)

is well defined. Observe that

Next we show that (γn ) is an increasing sequence, bounded above by the first eigenvalue. We will need the following lemma: Lemma 2.3. Let Ω ⊂ R N be a smooth bounded region and h ∈ C 0 (Ω) a nonnegative function. 1,p If u ∈ W0 (Ω) ∩ C 1,α (Ω) is a positive solution to the Dirichlet problem 

−p u = λp up−1 + h in Ω, u=0 on ∂Ω,

(14)

then h ≡ 0 in Ω and consequently u is a positive eigenfunction corresponding to the first eigenvalue λp . Proof. This proof is adapted from [1, Theorem 2.4] and based on the inequality  wp ∇u · ∇ p−1 , u 

|∇w|  |∇u| p

p−2

(15)

valid for all differentiable functions u, w in Ω that satisfy u > 0 and w  0; this inequality follows from Picone’s identity.

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Multiplying Eq. (14) by any v ∈ W0 (Ω) and integrating on Ω, we have 

 |∇u|p−2 ∇u · ∇v dx =

Ω

  λp up−1 + h v dx.

(16)

Ω

1,p

Let up ∈ W0 (Ω) ∩ C 1,α (Ω) be a positive eigenfunction corresponding to λp . Applying (15) with w = up and integrating on Ω, we find that 



 |∇up | dx 

|∇u|

p

Ω

p−2

p  up ∇u · ∇ p−1 dx. u

Ω 1,p

p

Now, since u > 0 in Ω, by Hopf’s lemma we have up /up−1 ∈ W0 (Ω). Therefore, we can apply (16) to conclude that 

 |∇up |p dx 

Ω

  upp λp up−1 + h p−1 dx. u

Ω

Thus,  0=

 |∇up |p dx −

Ω

which implies that h ≡ 0.



p

λp up dx 

h

Ω

φp dx  0, up−1

Ω

2

Proposition 2.4. For all n  2 the following hold: (i) γ0 < λp . (ii) γ0  γn  γn+1 < λp . (iii) There exists γ := lim γn and γ0  γ  λp . Proof. Property (iii) is a direct consequence of (i) and (ii). We prove (i) by a contradiction argument. Assume γ0  λp . Then, setting p−1

h = 1 − λp φ1

,

it follows by (13) that p−1

h  1 − γ0 φ1

 0.

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251

Write 

p−1

−p φ1 = 1 = λp φ1 φ1 = 0

+h

in Ω, on ∂Ω.

From Lemma 2.3, we conclude that h ≡ 0 and so λp φ1 ≡ 1. This implies that φ1 is constant, contradicting −p φ1 = 1 in Ω. In order to prove (ii), observe that γ0  γn follows immediately from Proposition 2.2: γ0 =



1 p−1

φ1 ∞



φn φn+1

p−1 .

The monotonicity of the sequence (γn ) can be shown by using the comparison principle. We have by definition 

p−1

p−1

−p φn = φn−1  γn−1 φn φn = 0 = φn+1 ,

 1/(p−1)  = −p γn−1 φn+1

in Ω, on ∂Ω,

whence 1/(p−1)

φn  γn−1

φn+1 ,

and, therefore   φn p−1  γn−1 . γn = inf Ω φn+1 Finally, in order to verify that γn < λp we use again a contradiction argument. Suppose that γn  λp for some n. Then,  p−1

p−1

λp φn+1  γn φn+1 

φn φn+1

p−1

p−1

p−1

φn+1 = φn

,

the second inequality a consequence of (12). Thus, p−1

− λp φn+1  0

p−1

= λp φn+1 + h

h := φn

p−1

in Ω. Since 

−p φn+1 = φn φn+1 = 0

p−1

in Ω, on ∂Ω,

it follows from Lemma 2.3 that h ≡ 0. Thus, p−1

φn

p−1

= λp φn+1

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and hence  λp ≡

φn φn+1

p−1

  φn p−1 = inf = γn . Ω φn+1

Furthermore, it also follows from (17) that  1/(p−1)  p−1 p−1 φn−1 = −p φn = −p λp φn+1 = λp (−p φn+1 ) = λp φn whence  λp ≡

φn−1 φn

p−1

  φn−1 p−1 = inf = γn−1 . Ω φn

Proceeding recursively we obtain λ p = γ0 , 2

which contradicts (i).

As a consequence from Propositions 2.2 and 2.4(i), we obtain the following behavior for the sequence (φn ): Corollary 2.5.

φn

φ1 n∞

→ 0 monotonically and uniformly.

Proof. Set wn :=

φn ,

φ1 n∞

so that, by (13), 

p−1

−p wn = γ0 wn−1 wn = 0

in Ω, on ∂Ω.

It follows from Proposition 2.2 that the sequence (wn ) is decreasing and uniformly bounded, since

φ1 n∞ φn+1 1 φn+1 wn+1 = = 1 n+1 φ wn

φ

φ1 ∞ n 1 ∞ φn and

wn ∞ =

φn ∞ φn−1 ∞ φ1 ∞ φn−2 ∞ φ1 2∞    · · ·  1.

φ1 n∞

φ1 n∞

φ1 n∞

The uniform boundedness of (wn ) together with the compacity of the operator (−p )−1 : C 0 (Ω) → C 0 (Ω) imply that (wn ) = (−p )−1 (γ0 wn−1 ) has a subsequence which converges uniformly to some function w ∈ C 0 (Ω). The monotonicity of (wn ) guarantees that the whole sequence converges uniformly and monotonically to w. The continuity of −−1 p implies that

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253

w = (−p )−1 (γ0 w). Therefore, 

−p w = γ0 w p−1 w=0

in Ω, on ∂Ω.

Since γ0 < λp and λp is the first eigenvalue for −p , we conclude that w = 0.

2

2.2. Sequence (Γn ) Set  Γn := sup Ω

φn φn+1

p−1

   φn p−1  = .  φ n+1 L∞ (Ω)

(18)

Observe that  p−1  φ0   = ∞, Γ0 =  φ  1 ∞ since φ0 = 1 on Ω and φ1 = 0 on the boundary ∂Ω. However, if one can guarantee that Γn0 is finite for some n0 , then sequence (Γn ) is well defined from n0 on: Proposition 2.6. Assume Γn0 < ∞ for some n0  1. Then 

φn φn+1

p−1  Γn0

for all n  n0

and therefore (Γn ) is well defined and bounded from above for n  n0 . Moreover, the sequence (Γn ) is decreasing for n  n0 . Proof. Proceeding by induction, assume we have shown Γn0 +k  · · ·  Γn0 . Then, for j = n0 +k we observe that    1  φj p−1 p−1 p−1 p−1 φj +1  Γj φj +1 = −p Γjp−1 φj +2 −p φj +1 = φj = φj +1 in Ω, and 1

φj +1 = 0 = Γjp−1 φj +2 on ∂Ω. Thus, it follows from the comparison principle that 1

φj +1  Γjp−1 φj +2

in Ω.

Hence,    φj +1 p−1    Γj . Γj +1 =  φj +2 ∞

2

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For special domains, we are able to prove that there exists n0 such that Γn0 is finite. Indeed, the following result holds: Proposition 2.7. Let Ω = BR be the ball of center at the origin and radius R. Then Γ1 is finite. Proof. Using the same notation as in Proposition 2.2, we have φn (x) = vn (r) with 1 1  θ  N −1 p−1 s p−1 vn−1 (s) ds dr. vn (r) = θ r

(19)

0

Thus, if x ∈ ∂BR , L’Hôpital’s rule implies that v  (r) φ1 (x) v1 (r) = lim = lim 1 = φ2 (x) r→R − v2 (r) r→R − v2 (r)



R

 1 N −1 ds p−1 0 s

R N −1 v (s)p−1 ds 1 0 s

< ∞.

2

2.3. Sequence (νn ) Set  νn =

φn Lp (Ω)

φn+1 Lp (Ω)

p−1 .

Clearly, sequence (νn ) is well defined. We show that both it and sequence (Γn ) are bounded below by the first eigenvalue. Proposition 2.8. There holds λp  νn  Γn for all n  1. Proof. By (4) and Hölder’s inequality we have    p−1  p p−1 p

∇φn+1 p = |∇φn+1 | dx = φn φn+1 dx  φn p φn+1 p , Ω

Ω

whence p

p−1

∇φn+1 p  φn p

φn+1 p .

Hence, from this and the variational characterization of the first eigenvalue λp = it follows that

inf

1,p v∈W0 (Ω)\{0}

∇v p ,

v p

R.J. Biezuner et al. / Journal of Functional Analysis 257 (2009) 243–270 p

λp 

∇φn+1 p p

φn+1 p

p−1



φn p

255

φn+1 p p

φn+1 p

= νn =

 

1 p−1

φn+1 p

Ω

φn φn+1

p

 p−1 p φn+1 dx

p

   p−1  p  φn p−1 1 p    φn+1 dx  p−1  φ n+1 ∞

φn+1 p Ω

= Γn .

2

Corollary 2.9. If lim Γn = λp then lim νn = λp . As an interesting consequence of Propositions 2.4 and 2.8 we have Corollary 2.10. If Ω is connected and Γn0 < ∞ for some n0  1, then for each n  n0 there exists at least one xn ∈ Ω such that λp =

φn (xn ) . φn+1 (xn )

3. The first eigenvalue and the first eigenfunction Recall that if γ = lim γn , then it follows from Proposition 2.4 that γ  λp . If we set Γ := lim Γn and ν := lim νn ,

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we have from Propositions 2.6 and 2.8 that ν < ∞ and λp  ν  Γ. We conjecture the following: Conjecture 3.1. There holds λp = γ = Γ = ν.

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In order to find a sequence of functions approximating the first eigenfunction, we define for each n ∈ N the function un :=

φn , an

where an is such that 1 an φn = γnp−1 = inf . Ω φn+1 an+1

For instance, if we set a1 := φ1 ∞ , then a2 =

a1

=

1/(p−1)

γ1

a1 infΩ

φ1 φ2

   φ2   = φ1 ∞  φ  1 ∞

and, in general, an =

φ1 ∞

1

φ1 φ2

infΩ

infΩ

φ2 φ3

···

1 infΩ

φn−1 φn

       φ2   φ3       · · ·  φn  . = φ1 ∞  φ  φ   φ 1 ∞ 2 ∞ n−1 ∞

Since    φk 

φk ∞  , 

φk−1 ∞  φk−1 ∞ we have an  φ1 ∞

φ2 ∞ φ3 ∞

φn ∞ ··· = φn ∞ .

φ1 ∞ φ2 ∞

φn−1 ∞

Therefore, un 

φn  1.

φn ∞

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257

1,p

Proposition 3.2. Let (un ) ⊂ W0 (Ω) ∩ C 1.α (Ω) be the sequence of functions defined above. Then (un ) is decreasing and satisfies 

p−1

−p un+1 = γn un un = 0

in Ω, on ∂Ω.

Furthermore, (un ) converges uniformly to a function u ∈ C 1.α (Ω) which satisfies 

−p u = γ up−1 u=0

in Ω, on ∂Ω.

Proof. We have  −p un+1 =

φn an+1

p−1

 =

an an+1

p−1

p−1

un

p−1

= γn un

.

Moreover, un+1 =

φn φn+1 φn+1 φn+1 φn φn = inf  = = un , an+1 an Ω φn+1 an φn+1 an

which proves that (un ) is decreasing and therefore we can define a function u in Ω by u(x) := lim un (x) for each x ∈ Ω. Since (un ) ⊂ C 1,α (Ω), 0  un  u1 , and the operator (−p )−1 : C 0 (Ω) → C 0 (Ω) is compact, the whole sequence (un ) converges to u uniformly and we can pass the limit in p−1

−p un+1 = γn un to obtain −p u = γ up−1 .

2

In view of Conjecture 3.1, this result suggests that u is the eigenfunction corresponding to the first eigenvalue. However, since we are not able to guarantee that u is not the null function, this does not follow automatically. On the other hand, since Proposition 3.2 is independent of the conjecture, its proof would show that γ = λp . We end this section by remarking that all results proved above are valid if we consider a positive weight ω(x) multiplying the right-hand side of the eigenvalue equation, that is, if we consider the equation −p u = λω(x)|u|p−1 u in Ω. The above arguments are easily adapted to contemplate this case and the most remarkable change appears in the sequence (νn ) which becomes  νn =

ω(x)1/p φn p

ω(x)1/p φn+1 p

p−1 .

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4. Spherical domains In this section we show the validity of Conjecture 3.1 for balls. Let B = B1 (0) ⊂ RN , N  2, denote the unit ball centered at the origin. The following lemma in the form it is given was first stated in [2], even though it has already often been used as a technical tool in differential geometry. A proof is provided for completeness. Lemma 4.1. Let f, g : [a, b] → R be continuous on [a, b] and differentiable in (a, b). Suppose  (x)−f (a) g  (x) = 0 for all x ∈ (a, b). If fg  is (strictly) increasing [decreasing], then both fg(x)−g(a) and f (x)−f (b) g(x)−g(b)

are (strictly) increasing [decreasing]. 

Proof. Assume fg  is increasing. Then the Cauchy mean value theorem implies that for each x ∈ (a, b) there exists y ∈ (a, x) such that f (x) − f (a) f  (y) f  (x) =    . g(x) − g(a) g (y) g (x) On the other hand, since g  = 0 we always have g  (x) > 0. g(x) − g(a) Thus,   f  (x) g  (x) f (x) − f (a) d f (x) − f (a) = − dx g(x) − g(a) g(x) − g(a) g(x) − g(a) g(x) − g(a)  and so If

f (x)−f (a) g(x)−g(a)

f g

g  (x) f  (x) f  (x) − = 0, g(x) − g(a) g(x) − g(a) g  (x)

is increasing. f (x)−f (a) g(x)−g(a)

is decreasing, then the same arguments prove that

the above inequalities are strict if the monotonicity of similar. 2

f g

is decreasing. Moreover,

is strict. The proof for

We use this lemma in order to show that for each n  0 the quotient function of r = |x|:

φn φn+1

Theorem 4.2. Let p > 1 and for r ∈ [0, 1] set φ0 (r) ≡ 1, 1  θ  φn (r) = r

0

s θ

N −1

p−1 φn−1 (s) ds

1 p−1

dθ,

if n  1.

f (x)−f (b) g(x)−g(b)

is

is increasing as a

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259

Then, for each n  1 the function φn is strictly decreasing and for each n  0 the quotient is strictly increasing on [0, 1].

φn φn+1

Proof. As  r   1 p−1 N −1 s p−1 φn (r) = − φn−1 (s) ds < 0, r 0

for r > 0, the functions φn are strictly decreasing for n  1. n In particular, the quotient φφn+1 is strictly increasing when n = 0. In order to show that the quotients are strictly increasing for n  1, we use an induction argument. Assume that the quotient φn−1 φn is strictly increasing for some n  1. Noticing that φn (1) = φn+1 (1) = 0, we can write φn (r) − φn (1) φn . (r) = φn+1 φn+1 (r) − φn+1 (1) We will apply the previous lemma in order to show that the quotient in the right-hand side of this equation is strictly increasing. For this, it is enough to verify that φn (r) =  φn+1 (r)

 r

1 N −1 φ p−1 (s) ds  p−1 n−1 0 s

r N −1 φ p−1 (s) ds n 0 s

1

is increasing. Since the map ξ → ξ p−1 is increasing, this is equivalent to showing that

r 0

s N −1 φn−1 (s) ds

0

s N −1 φn

r

p−1 p−1

(s) ds

is increasing. But this is itself a consequence of the lemma, for both

r N −1 p−1 φn (s) ds equal zero at r = 0 and 0 s

r   p−1 ( 0 s N −1 φn−1 (s) ds) φn−1 (r) p−1 = ,

r p−1 φn (r) ( 0 s N −1 φn (s) ds) which is strictly increasing by the induction hypothesis.

2

Corollary 4.3.     φn p−1

φn ∞ p−1 γn = inf = . B φn+1

φn+1 ∞

r 0

s N −1 φn−1 (s) ds and p−1

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Proof. Since φn and φn+1 are decreasing functions, we have φn ∞ = φn (0) and φn+1 ∞ = n is increasing, φn+1 (0). Therefore, since φφn+1       φn p−1 φn (0) p−1

φn ∞ p−1 inf = = . B φn+1 φn+1 (0)

φn+1 ∞

2

Theorem 4.4. Let (un ) be the sequence defined by un :=

φn

φn

for each n ∈ N. Then γ = λp (B) and (un ) converges uniformly (and monotonically) to a positive function u ∈ C 1,α (B) such that

u ∞ = 1 and  −p u = λp up−1 in B, (22) u=0 on ∂B. Proof. Notice that sequence (un ) is the same as that defined in Proposition 3.2, since an defined in (21) is now, in view of Corollary 4.3, an =

φ1 ∞

1

infΩ φφ12

infΩ φφ23

···

1 infΩ φφn−1 n

=

φ1 ∞

1

φ1 ∞ φ2 ∞

φ2 ∞ φ3 ∞

Thus, it satisfies the nonlinear problem  p−1 −p un+1 = γn un un = 0

···

in B, on ∂B,

1

φn−1 ∞

φn ∞

= φn ∞ .

(23)

and is decreasing. Therefore, arguing as in the proof of Proposition 3.2, we can pass the limit in (23) in order to obtain (22). However, differently from the sequence of Proposition 3.2, we have in addition

un ∞ = 1 for every n ∈ N, which allows us to conclude that u ∞ = lim un ∞ = 1, hence u is not the null function and thus γ = λp (see remark after Proposition 3.2). 2 Next we will show that the sequence (φn /φn+1 ) converges uniformly to the constant function λp on each compact set contained in B. Lemma 4.5. For each 0 < ε < 1 define  1   N−1 −1 ε p−1 Kε := dθ . θ 1−ε

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261

Then  0

φn φn+1



   φ1    Kε  φ  2 ∞

on the interval [ε, 1 − ε].

(24)

Proof. Since (see Propositions 2.6 and 2.7)   φn  φ

n+1

   

   φ1    φ  , 2 ∞ ∞

if n  2

(25)

and  0

φn φn+1



=

 φn+1 φn − φn φn+1 2 φn+1

=

  | |φn | | φn |φn+1 φn |φn+1 −  , φn+1 φn+1 φn φn+1 φn+1

it suffices to show that  |φn+1 |

φn+1

 Kε

in [ε, 1 − ε].

And indeed, for ε  r  1 − ε < 1 we have 1 1  θ  N −1 p−1 s p−1 φn (s) ds dθ φn+1 (r) = θ r

0

 1 

θ

− N−1 p−1

 r dθ

r

p−1 s N −1 φn (s) ds

1 p−1

0

 1 =



θ

− N−1 p−1

N−1

 dθ r p−1 φn+1 (r)

1−ε

 1   N−1



ε p−1  dθ φn+1 (r) . θ

2

1−ε

Theorem 4.6. For each fixed 0 < ε < 1 we have 

φn (|x|) φn+1 (|x|)



1 p−1

→ λp

uniformly on the annulus Ωε1−ε := {ε < |x| < 1 − ε} ⊂ B1 . Proof. In view of (24) and (25), it follows from Arzela–Ascoli’s theorem that, up to a subsen (r) quence, ( φφn+1 (r) ) converges uniformly to a function w ∈ C([ε, 1 − ε]).

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Taking un (|x|) =

φn (|x|)

φn ∞

as in the proof of Theorem 4.4, we can write 

p−1

−p un+1 =

φn

p−1

φn+1 ∞

=

φn φn+1

p−1

p−1

in Ωε1−ε .

un+1

Thus, passing the limit in this equation for a convenient subsequence, we obtain −p u = w p−1 u, where u is the eigenfunction given in Theorem 4.4. Therefore, w p−1 ≡ λp , for −p u = λp up−1 . The proof is complete since the limit is independent of the particular subsequence that converges to w. 2 Corollary 4.7. Γ = λp . Proof. We have  Γn = sup 0r1

φn φn+1

p−1

 = lim

r→1+

φn (r) φn+1 (r)

p−1

 =

φn (1)  φn+1 (1)

p−1

and 

φn (1)  φn+1 (1)

1

p−1 =

N −1 φ p−1 (s) ds n−1 0 s

1 N −1 φ p−1 (s) ds s n 0

1

=

0

φn (s) p−1 s N −1 ( φφn−1 (s))p−1 ( φ ) ds n n ∞ .

1 N −1 ( φn (s) )p−1 ds s 0

φn ∞

and φφnn ∞ are bounded, λp = lim( φφn−1 )p−1 and u(|x|) = lim( φ φn (|x|) )p−1 is the eigenSince φφn−1 n n n ∞ function obtained above, we can apply Lebesgue’s dominated convergence theorem to obtain 

φ  (1) Γ = lim  n φn+1 (1)

1

p−1 =

0

s N −1 λp up−1 (s) ds = λp .

1 N −1 up−1 (s) ds 0 s p−1

Corollary 4.8.  

φn p p−1 ν := lim = λp .

φn+1 p

2

R.J. Biezuner et al. / Journal of Functional Analysis 257 (2009) 243–270

Proof. In view of the last corollary, this follows from Corollary 2.9.

263

2

This result in fact holds for any Lq -norm (q > 1): Corollary 4.9. For any q > 1 there holds  

φn q p−1 lim = λp .

φn+1 q Proof. As before set u(|x|) = lim( φ φn (|x|) )p−1 . Then it follows from Lebesgue’s dominated conn ∞ vergence theorem that    1 N −1 q  p−1 q φn (s) ds

φn q p−1 0 s lim = lim 1 q

φn+1 q s N −1 φ (s) ds n+1

0

  1 s N −1 lim( φn (s) )q ds  p−1  q

φn ∞ p−1 0

φn ∞ = lim

1

φn+1 ∞ s N −1 lim( φn+1 (s) )q ds 0

 1 = λp 01 0

= λp .

s N −1 uq (s) ds



φn+1 ∞

p−1 q

s N −1 uq (s) ds 2

As at the end of last section, we also remark that all results in this section remain valid if we consider a radially symmetric weight ω(|x|). Moreover, the results can be naturally extended to the Dirichlet problem in RN with an appropriate weight. 5. The case p = 2 In this section, we give a complete proof of the convergence of the three sequences to the first eigenvalue of the Laplacian. Let 0 < λ1 < λ2  λ3  · · · be the increasing sequence of Dirichlet eigenvalues for the Laplacian − in Ω and (en )n∈N ⊂ W01,2 (Ω) ∩ C 2 (Ω) be a corresponding sequence of eigenfunctions which is also an orthogonal system for L2 (Ω) and normalized by the sup-norm, that is, en ∞ = 1 for all n ∈ N. Denote the inner product in L2 (Ω) by  u, v =

uv dx. Ω

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If ξ ∈ W01,2 (Ω) ∩ C 2 (Ω) is such that ξ > 0 in Ω, then ξ=

∞ 

αk ek

k=1

with αk = ξ, ek ,

k = 1, 2, . . . ,

and we may assume  α1 = ξ, e1  =

ξ e1 dx > 0, Ω

since we can take e1 > 0 in Ω. Moreover, ∞ 

αk2 ek 2 = ξ 22 − α12 e1 2 < ξ 22 .

k=2

Now, if φ ∈ W01,2 (Ω) ∩ C 2 (Ω) is such that 

−φ = ξ φ=0

in Ω, on ∂Ω,

it follows that ∞  φ= φ, ek ek k=1

with ∞ 

αk ek = ξ = −φ =

k=1

∞  φ, ek (−ek ) k=1

=

∞ 

λk φ, ek ek ,

k=1

whence φ, ek  =

αk . λk

Thus, φ=

∞  αk k=1

λk

ek .

(26)

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265

Returning to sequence (φn ), if 1=

∞ 

αk ek

k=1

is the expansion of the function ξ ≡ 1, we obtain recursively φn =

∞  αk k=1

λnk

ek ,

that is, φn =

1 (α1 e1 + ψn ), λn1

where ψn :=

 ∞   λ1 n λk

k=2

αk ek .

n 2 Theorem 5.1. λ1 = lim φ φn+1

2 .

Proof. We assert that if (ψn )n∈N is the sequence defined above, then ψn → 0 in L2 (Ω). Indeed, this follows immediately from the estimate

ψn 22 =

 ∞   λ1 2n k=2

λk

 αk2 ek 22 

λ1 λ2

2n  ∞ k=2

 αk2 ek 22  1 2

λ1 λ2

n

and the fact that λ1 < λ2 . Therefore, lim

 1   2 α e1 2 + ψn 22 2

φn 2

α1 e1 + ψn 2 = λ1 lim 2 1 2 2 = lim λ1 = λ1 .

φn+1 2

α1 e1 + ψn+1 2 α1 e1 2 + ψn+1 22

Theorem 5.2. ψn → 0 uniformly in Ω. Proof. Since the convergence of the eigenfunction expansion of ξ ≡ 1 is absolute, let M :=

∞  k=1

|αk |.

2

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Then  ∞    λ n 1  

ψn ∞ =  αk ek    λk

∞

k=2

and the result follows letting n → ∞. Corollary 5.3.

φn φn+1

 M

λ1 λ2

n

2

→ λ1 uniformly in any compact subset K  Ω.

Proof. Let K  Ω be a compact set and set mK := min e1 > 0. K

Since ψn → 0 uniformly in Ω, we have for all sufficiently large n that α1 mK  α1 e1  |α1 e1 + ψn+1 | + |ψn+1 | α1 < |α1 e1 + ψn+1 | + mK , 2 whence |α1 e1 + ψn+1 | 

α1 mK . 2

Thus, on K we obtain





φn

α1 e1 + ψn





− λ1 = λ1

− 1

φ α1 e1 + ψn+1 n+1



ψn − ψn+1

= λ1



α e +ψ 1 1

n+1

2λ1  |ψn − ψn+1 | α1 mK →0 uniformly.

2

6. Higher eigenvalues In the case p = 2, higher eigenvalues and their respective eigenfunctions can also in principle be obtained by this technique. Suppose now that the first nonzero coefficient of ξ ∈ W01,2 (Ω) ∩ C 2 (Ω) is αk0 for some k0 > 1, that is, ξ=

∞  k=k0

αk ek

(27)

R.J. Biezuner et al. / Journal of Functional Analysis 257 (2009) 243–270

267

with αk = v, ek ,

k = k0 , k0 + 1, . . . .

Then φn =

∞  αk ek , λnk

k=k0

which we write in the form φn =

1 (αk0 ek0 + ψn ), λnk0

(28)

where now ψn :=

 ∞   λk0 n αk ek . λk

k=k0 +1

Following the same steps of the previous section, we can conclude that ψn → 0 in L2 (Ω) and that λk0 = lim

φn 2 .

φn+1 2

Moreover, choosing ξ sufficiently regular so that the series of the coefficients M :=

∞ 

|αk |

k=1

is absolutely convergent, we also have   ∞     λ n   k0 αk ek 

ψn ∞ =    λk k=k0 +1

∞

 M

λk0 λk0 +1

n ,

n that is, ψn → 0 uniformly in Ω, which implies as above that φφn+1 converges uniformly to the constant function λk0 in compact subsets of K  Ω ∩ supp(ek0 ).

7. Numerical results In this section we present some of the numerical results which we were able to compute for some domains. We compare them with results obtained elsewhere. Computations were performed on a Windows XP/Pentium 4 – 2.8 GHz platform, using the GCC compiler.

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Table 1 First eigenvalue for the p-Laplacian on the unit ball. p

N =2

N =3

N =4

p

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5

2.5694 2.9656 3.3263 3.6741 4.0180 4.3624 4.7098 5.0619 5.4195 5.7835 6.1543 6.5321 6.9174 7.3103 7.7108

3.8728 4.5151 5.1283 5.7431 6.3717 7.0201 7.6920 8.3898 9.1153 9.8698 10.6545 11.4701 12.3177 13.1979 14.1115

5.1871 6.1020 7.0064 7.9390 8.9154 9.9443 11.0314 12.1810 13.3969 14.6822 16.0400 17.4730 18.9841 20.5759 22.2510

2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

N =2 8.1192 8.5355 8.9598 9.3921 9.8324 10.2809 10.7375 11.2022 11.6751 12.1561 12.6453 13.1427 13.6482 14.1619 14.6838

N =3

N =4

15.0590 16.0412 17.0586 18.1117 19.2013 20.3278 21.4917 22.6937 23.9341 25.2136 26.5327 27.8919 29.2916 30.7325 32.2150

24.0121 25.8617 27.8027 29.8374 31.9687 34.1991 36.5314 38.9681 41.5120 44.1659 46.9325 49.8144 52.8146 55.9359 59.1810

Fig. 1. Graphs of p (1 < p  4) versus values of γ10 , ν10 and Γ10 for the N -dimensional unit ball and N = 2 (left), N = 3 (center), N = 4 (right).

7.1. The unit ball In order to compute the value of the first eigenvalue for the p-Laplacian in the unit ball, we mixed the composite Simpson and trapezoidal methods for computation of the associated integrals in the expression of νn . In Table 1, the results for the first eigenvalue of the p-Laplacian for values of p ranging from 1.1 to 4.0 for balls of dimensions N = 2, 3, 4, are displayed, truncated at the fourth decimal place, after 10 iterations. The results are also visually displayed in Fig. 1. For comparison, the known value of the first eigenvalue for the Laplacian on the unit bidimensional ball is 5.7832, which means that our error should be about 0.04%. This result compares well with the one obtained in [8], where a 1.3% precision was attained. 7.2. The unit square In order to solve the p-Laplacian in the unit square [0, 1] × [0, 1] we used the algorithm proposed in [3], coupled with the homotopy perturbation method (HPM) of [6] for the exact line searches in the nonlinear conjugate gradient method. In Table 2 we see the values for the first eigenvalue of the p-Laplacian for values of p ranging from p = 2 to p = 3,

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269

Table 2 First eigenvalue for the p-Laplacian on the unit square. p

γ5

ν5

Γ5

2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

19.7145 22.3239 25.2168 28.2413 31.9750 35.5746 38.5547 41.4917 5.5593 7.8823 14.6719

19.7348 22.3460 25.2412 28.4495 32.0024 35.9344 40.2827 45.0890 50.3972 56.2567 62.7208

19.9270 22.4447 25.3343 28.6139 32.5685 37.6961 40.8167 52.8657 642.6432 670.7254 205.0535

Fig. 2. Graph of p (2  p  3) versus ν5 for the square [0, 1] × [0, 1].

with 0.1 increment, truncated at the fourth decimal place, after 5 iterations, for all three sequences. We see that sequence (νn ) has a much faster rate of convergence and less numerical error, especially as the value of p increases. For comparison, the known value of the first eigenvalue for the Laplacian on the unit square is 2π 2 = 19.7392, which means that the error in ν5 is about 0.02%. Fig. 2 displays the results for ν5 . This result with only 5 iterations compares very favourably with the one obtained in [8], where a 3% precision was attained. It must be remarked that better and faster results should be obtainable with more precise and faster methods for solving the p-Laplacian equation in each iteration.

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Acknowledgment The second author thanks the support of FAPEMIG. References [1] W. Allegretto, Y.X. Huang, A Picone’s identity for the p-Laplacian and applications, Nonlinear Anal. 32 (1998) 819–830. [2] G.D. Anderson, M.K. Vamanamurthy, M. Vuorinen, Inequalities for quasiconformal mappings in space, Pacific J. Math. 160 (1993) 1–18. [3] B. Andreianov, F. Boyer, F. Hubert, Finite volume schemes for the p-Laplacian on Cartesian meshes, ESAIM Math. Model. Numer. Anal. 38 (6) (2004) 931–959. [4] L. Damascelli, Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results, Ann. Sc. Norm. Super. Pisa Cl. Sci. 26 (4) (1998) 689–707. [5] J.W. Demmel, Applied Numerical Linear Algebra, SIAM, 1997. [6] X. Feng, Y. He, High order iterative methods without derivatives for solving nonlinear equations, Appl. Math. Comput. 186 (2007) 1617–1623. [7] B. Kawohl, V. Fridman, Isoperimetric estimates for the first eigenvalue of the p-Laplace operator and the Cheeger constant, Comment. Math. Univ. Carolin. 44 (2003) 659–667. [8] L. Lefton, D. Wei, Numerical approximation of the first eigenpair of the p-Laplacian using finite elements and the penalty method, Numer. Funct. Anal. Optim. 18 (3–4) (1997) 389–399. [9] D.S. Watkins, Fundamentals of Matrix Computations, second ed., John Wiley & Sons, 2002.

Journal of Functional Analysis 257 (2009) 271–331 www.elsevier.com/locate/jfa

Isomorphic copies in the lattice E and its symmetrization E (∗) with applications to Orlicz–Lorentz spaces Anna Kami´nska a , Yves Raynaud b,∗ a Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, United States b Institut de Mathématiques de Jussieu (UMR 7586 CNRS), Case 186, UPMC University Paris 06,

F-75005 Paris, France Received 30 October 2008; accepted 25 February 2009 Available online 9 April 2009 Communicated by N. Kalton

Abstract The paper is devoted to the isomorphic structure of symmetrizations of quasi-Banach ideal function or sequence lattices. The symmetrization E (∗) of a quasi-Banach ideal lattice E of measurable functions on I = (0, a), 0 < a  ∞, or I = N, consists of all functions with decreasing rearrangement belonging to E. For an order continuous E we show that every subsymmetric basic sequence in E (∗) which converges to zero in measure is equivalent to another one in the cone of positive decreasing elements in E, and conversely. Among several consequences we show that, provided E is order continuous with Fatou property, E (∗) contains an order isomorphic copy of p if and only if either E contains a normalized p -basic sequence which converges to zero in measure, or E (∗) contains the function t −1/p . We apply these results to the family of two-weighted Orlicz–Lorentz spaces Λϕ,w,v (I ) defined on I = N or I = (0, a), 0 < a  ∞. This family contains usual Orlicz–Lorentz spaces Λϕ,w (I ) when v ≡ 1 and Orlicz–Marcinkiewicz spaces Mϕ,w (I ) when v = 1/w. We show that for a large class of weights w, v, it is equivalent for the space Λϕ,w,v (0, 1), and for the non-weighted Orlicz space Lϕ (0, 1) to contain a given sequential Orlicz space hψ isomorphically as a sublattice in their respective order continuous parts. We provide a complete characterization of order isomorphic copies of p in these spaces over (0, 1) or N exclusively in terms of the indices of ϕ. If I = (0, ∞) we show that the set of exponents p for which p lattice embeds in the order continuous part of Λϕ,w,v (I ) is the union of three intervals determined respectively by the indices of ϕ and by the condition that the function t −1/p belongs to the space.

* Corresponding author.

E-mail addresses: [email protected] (A. Kami´nska), [email protected] (Y. Raynaud). 0022-1236/$ – see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2009.02.016

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© 2009 Elsevier Inc. All rights reserved. Keywords: Quasi-Banach lattice of measurable functions; Symmetrization; Rearrangement invariant spaces; Orlicz–Lorentz spaces; Isomorphic lattice structure

We present a comprehensive study of (order) isomorphic copies of p for 0 < p  ∞ as well as c0 in symmetrized quasi-Banach ideal (function) lattices. Given a function lattice E over I = N or I = (0, a), 0 < a  ∞, and the quasi-norm  · E in E, define its symmetrization E (∗) as the set of all elements x in E such that their decreasing rearrangements x ∗ belong to E, and let xE (∗) = x ∗ E . The continuity of the dilation operator in E guarantees that the latter functional is also a quasi-norm. There are a number of important examples of symmetric spaces that are defined by this procedure. The most important are all sorts of Lorentz spaces, including classical Lp,q , 0 < p, q < ∞, Λp,w with a positive weight w and 0 < p < ∞, Orlicz–Lorentz spaces Λϕ,w with ϕ being an Orlicz function, and the Marcinkiewicz spaces Mp,w or more general Orlicz–Marcinkiewicz spaces Mϕ,w , being in general dual spaces to Lorentz spaces. While studying the problem of isomorphic copies in different spaces, including in particular Orlicz– Lorentz spaces [23,24], we realized that some general methods could be applied to all spaces of the type E (∗) . We present here a general approach to investigate asymptotic relationships between certain type of basic sequences in E and E (∗) . For the cases we have in mind, the study of basic disjoint sequences is far easier in E than in E (∗) . This allows us to apply these schemes for different spaces E without repeating similar reasonings. The paper is organized as follows. The first section contains preliminaries such as basic notions, notations as well as basic facts. For instance we show that the continuity of the dilation operator on a complete quasi-normed space E is necessary and sufficient for the symmetrization E (∗) to be a quasi-normed complete space. In Section 2 we discuss a number of relationships among basic sequences in the spaces E (∗) and E (∗) . Let us denote by E0 the order continuous part of E (∗) and by Eb the closure in E of the linear subspace of bounded elements with supports in finite intervals. We suppose that E0(∗) = {0}. We prove first a key result on equivalence between a finite or infinite disjoint sequence in E (∗) consisting of bounded elements supported on sets of finite measure, and some sequence of suitable translations of their rearrangements in E. The equivalence constant is controlled in a precise way by a certain function of the E (∗) -norm and L∞ -norm of the elements, and of the amplitudes of the shifts. This technical result leads to the main result of the section, which states that every seminormalized sequence in E (∗) which converges to zero in measure (“L0 -null” in short) has a basic subsequence equivalent to a L0 -null semi-normalized sequence in Eb , consisting of non-negative decreasing elements. Conversely such a sequence in Eb has a basic subsequence equivalent to a (∗) semi-normalized L0 -null sequence in E0 . An immediate consequence of this result is that every subsymmetric semi-normalized L0 -null basic sequence in E (∗) is equivalent to a (subsymmetric) semi-normalized L0 -null basic sequence in Eb , consisting of non-negative decreasing elements, and conversely. Note that by standard disjointification arguments, every semi-normalized L0 -null sequence (∗) in E0 has a basic subsequence close to and equivalent to a disjoint L0 -null sequence, and we show that the same happens in Eb for sequences of decreasing non-negative elements. Thus

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every subsymmetric, L0 -null, semi-normalized and disjoint sequence in E0 is equivalent to a sequence in Eb with the same properties, but now the converse may be false. These results are the main tools for further consideration of isomorphic structure of the spaces. In fact, this method often shifts the complicated constructions from the space E (∗) to the space E, which makes them considerably easier since it does not require dealing with decreasing rearrangements. In Section 3 we state general necessary and sufficient conditions for the existence of subspaces or sublattices isomorphic to p , 0 < p < ∞, or c0 , in order continuous rearrangement invariant quasi-Banach spaces. The general form of these conditions is that either the r.i. space X contains a normalized L0 -null basic sequence equivalent to the unit basis of p (or c0 ), or X contains certain peculiar decreasing functions. This kind of criterion is well adapted to the case where X (∗) is the order continuous part E0 of the symmetrization of an ideal function space E. In fact by the results of Section 2, these conditions transfer to analogous conditions on the cone of decreasing non-negative elements in E. The outline of the first criterion on p -subspaces is that an order continuous rearrangement invariant space X contains p isomorphically as subspace if and only if either it contains p isomorphically as sublattice, or p  2 and the “Fatou closure” of its restriction to [0, 1] contains p-stable symmetric random variables, which can be replaced by the function t −1/p if p < 2. This is an extension to the case of quasi-Banach spaces of a known result presented in [36]. We only give a sketch of its proof, attaching a result on p subspaces in Lr , 0 < r  1, in the spirit of Aldous [1] and Dacunha-Castelle and Krivine [8], in the appendix in the last section. We then analyze the existence of order isomorphic copies of p or c0 . In the case of sequence spaces (different from c0 ) or of spaces over finite intervals, the existence of a sublattice isomorphic to p or c0 implies the existence of an L0 -null normalized basic sequence, which spans the sublattice and is equivalent to the p or c0 unit basis. In the spaces over (0, ∞), there are two fundamental types of p or c0 isomorphic sublattices. The first ones are spanned by a normalized basic sequence converging to zero in measure, the second ones are spanned by a disjoint basic sequence of elements, the rearrangements of which converge pointwise to the function t −1/p . In fact, if an p or c0 isomorphic sublattice exists, there must exist a sublattice of one of these two types. The existence of the second type of p -sublattice is equivalent to the fact that the order part L0p,∞ of the Marcinkiewicz space Lp,∞ is included in X. If X has Fatou property it is equivalent to the inclusion Lp,∞ ⊂ X, and also to the fact that X contains the function t −1/p . Our reasoning here is based on a recent paper by Hernandez, Sanchez and Semenov [13]. It needs a supplementary technical hypothesis on the r.i. space, namely that this quasi-Banach lattice X is L-convex, i.e. it is r-convex for some 0 < r < ∞. We do not know if this additional hypothesis can be removed. Finally, for a function space E we give criteria for the containment of p or c0 in E (∗) as a subspace or sublattice in terms of conditions imposed on the space E. They vary depending on the underlying set I . The rest of the paper consists of investigations of the isomorphic structure of a new, very general class of Orlicz–Lorentz spaces, so called two-weighted Orlicz–Lorentz spaces Λϕ,w,v (I ). The techniques and results developed in sections two and three find very natural applications for studying basic sequences in these spaces. Given two positive weights w, v, and an Orlicz function ϕ (not necessarily convex), we introduce in Section 4 the space Λϕ,w,v (I ) as the set of all measurable functions x : I → R such that for some λ > 0,    ϕ λx ∗ (t)v(t) w(t) dt < ∞ I

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(in the sequential case, i.e. when I = N, this space will also be denoted by d(w, v, ϕ), consistently with the traditional notation in the classical case, see [27]). Clearly, these spaces are obtained by symmetrization of certain two-weighted Orlicz spaces that belong to the more general family of Musielak–Orlicz spaces. The new family of spaces Λϕ,w,v (I ) is very vast and contains not only all sorts of classical Lorentz and Orlicz–Lorentz function and sequence spaces but also in most cases their dual spaces, known as Marcinkiewicz or Orlicz–Marcinkiewicz spaces [15,22,37]. In fact such an Orlicz–Marcinkiewicz space Mϕ,w (I ) is just a space Λϕ,w,v (I ) with v = 1/w. This latter fact was our prime motivation for working with the family of twoweighted spaces. Many results for Marcinkiewicz spaces are obtained for granted as corollaries of our main theorems. Note also that the spaces considered by Torchinsky [39], which appear naturally as real interpolation spaces [35], belong also to this class with v increasing and w(t) = 1/t. We provide in Section 4 a set of sufficient conditions for both weights w, v and the Orlicz function ϕ in order our space to become a quasi-Banach space. We also state several lemmas and auxiliary facts that connect different properties of the weights and the function ϕ. The next Section 5 contains criteria for Λϕ,w,v (I ) to have (order) isomorphic copies of c0 or ∞ , and some preparatory work needed further. After recalling the sets of Orlicz functions classically attached to a given Orlicz function ϕ, and used for studying subspaces and sublattices of different types of Orlicz spaces, we finish this section with a general theorem stating that Λϕ,w,v (I ), independently of the sort of I , needs to contain a sublattice isomorphic to p for some 0 < p < ∞. More precisely, in the order continuous part Λ0ϕ,w,v (I ) of the space, the closed linear span of any semi-normalized sequence converging in measure to zero contains a subsequence equivalent to the unit basis of some p , which is also equivalent to a disjoint sequence in the Orlicz space Lϕ (0, ∞). Recall that the characterization of sublattices of Orlicz spaces Lϕ (I ), which are isomorphic to spaces p , depends strongly on the interval I . This was demonstrated in early papers by Lindenstrauss and Tzafriri [27,29], by Nielsen [33], and Hernandez with Rodriguez-Salinas [12], where it was also pointed out that the strongest differences occur between spaces defined over finite and infinite interval I . At this point several natural questions arise: (a) Does every lattice copy of p in Λ0ϕ,w,v (I ) have an order copy in Lϕ (0, ∞)? (b) Does every lattice copy of p in Λ0ϕ,w,v (I ) have an order copy in Lϕ (I ) for the same I ? (c) Conversely, is every lattice copy of p in Lϕ (I ) isomorphic to a sublattice of Λ0ϕ,w,v (I )? Recall that the answers to these questions are positive in the case of I = N and a constant weight v [24]. It turns out that when v is constant the answers are still positive for a finite interval I = (0, a), but when either I = (0, ∞) or v is not constant, the situation is far more diversified. Section 6 is devoted to spaces over finite intervals (0, a). In this case we are able to obtain a positive answer to question (b), and hence (a) under the assumptions that the weight w is integrable over every finite interval (0, t), and to question (c) under the additional condition that the weight v is increasing. Thus under all these assumptions together we provide a complete characterization of all sublattices of Λϕ,w,v (0, a) isomorphic to p , 0 < p  ∞ (Theorem 6.13). These conditions are expressed in terms of Matuszewska–Orlicz indices similarly as in [33]. Actually we obtain a more general result, namely that an Orlicz sequence space hψ embeds isomorphically as sublattice into Λ0ϕ,w,v (0, a) if and only if it does into L0ϕ (0, a). Moreover we show that every infinite dimensional sublattice of Λϕ,w,v (0, a) contains an order copy of some space p .

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In Section 7 we treat the case when the measure of I is infinite, that is when either I = N or I = (0, ∞). In either case there is a similar situation when considering normalized disjoint sequences approaching zero in the ∞-norm. Under certain mild conditions on the weights v, w such sequences have a subsequence spanning an Orlicz space hψ , which is also isomorphic to a sublattice of hϕ . In the case of sequence spaces, under the same conditions on the weights, we obtain a positive answer to questions (a) and (b), and assuming in addition that v is increasing, we also get a positive answer to question (c). Under these assumptions, in Theorem 7.8, we determine that p -spaces embed isomorphically in d0 (w, v, ϕ), the order continuous part of d(w, v, ϕ), if and only if they embed in hϕ . Note however that the equivalence of order embeddings of hψ in d0 (ϕ, w, v) and in hϕ is obtained now only for a restricted class of embeddings, those which associate the hψ basis with a ∞ -null seminormalized disjoint sequence. As an application we show that a reflexive Orlicz–Lorentz sequence space with decreasing weight contains a subspace isomorphic to p for some p  1 if and only if the space has a quotient space isomorphic to p , analogously to reflexive Orlicz spaces [27]. As we can expect in view of the general results in Section 3, the case when I = (0, ∞) is more complicated. This is caused by the fact that there may exist subspaces isomorphic to p that are spanned by normalized sequences of equimeasurable functions. Our main result, Theorem 7.18, states that p is lattice embeddable into Λ0ϕ,w,v (0, ∞) if and only if either it is isomorphic to a sublattice of hϕ or Lϕ (0, 1), or the function t −1/p belongs to Λϕ,w,v (0, ∞). This determines three intervals in the positive real line, the first two depending only on the function ϕ, while the third one depends also on the weights. The latter condition means that Lp,∞ ⊂ Λϕ,w,v (0, ∞). We also show, inspired by Carothers and Dilworth’s paper [6], that under the assumption