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 ⇑ Sequentia