Banach algebras: An introduction (Pure and Applied Mathematics: A Series of Monographs and Textbooks 24)

107 compact Abelian topological group G is the counting measure. particular this means that A((SJ) = 1, s E G.

In

Theorem 4.7.1. Let G be a locally compact Abelian topological group. Then the following are equivalent: Ci) (ii)

LI(G)

is a commutative Banach algebra with identity.

G is discrete.

Proof. Suppose G is discrete and define e(t) = 1, for t = 0, and e(t) = 0, for t ~ 0, have used 0 to denote the additive identity in An easy computation then reveals that e * f = f whence e is an identity for LI(G). Thus part implies part (i).

e (Ll(G) by t E G, where we both C and G. * e = f, f E L1(G), (ii) of the theorem

Conversely, suppose Ll(G) has an identity e. we claim that there exists some p > 0 such that inf{A(O)

I

0 c G, 0

~ ~,

In this case

0 open) > p.

Indeed, suppose this were not the case and let e ~ Ll(G), it is apparent that the formula

0 < e < 1.

Since

~(E) = IE le(t)1 dA(t),

where E is a Borel subset of G, defines a regular Borel measure on G that is absolutely continuous with respect to A. Thus there exists some 6 > 0 such that, whenever E eGis a Borel set for which A(E) < 0, then ~(E) < e. In particular there exists an open neighborhood U of 0 in G such that

Su

le(t)1 dA(t) < e.

This latter assertion is valid as we are assuming that there exist open subsets of G with arbitrarily small Haar measure and because Haar measure is translation invariant.

108

4. The Gel'fand Representation of Specific Algebras

However, from the continuity of the group operations in G there exists an open neighborhood W of 0 in G such that W= -w and W+ WC U. Denoting the characteristic function of W by Xw, we see that Xw(s) =

~,

* e(s)

=

IG

Xw(s - t)e(t) dAft)

=

fw+s Xw(s - t)e(t) dA(t)

Consequently, if sEW, 1

= Xw(s)

(s E G).

then ~ Iw+s ~~(s - t)e(t) dAft)

< Iw+w le(t)1 dAft)

~Iu le(t)1 dAft) < e,

contrary to the choice of e. that inf{A(O)

I0c

Hence there exists some

p

> 0 such

G, 0 ~ ~, 0 open) > p.

But now suppose that 0 eGis open and that 0 has compact closure. The regularity of Haar measure entails that ACO) <~. If o contains n distinct points, then clearly there exist pairwise disjoint open neighborhoods of these n points that are contained in O. Call them 01,02, ... ,On0 Then n n ACO) > A( U Ok) = t A(Ok) > np. k=l k=l Since A(O) < m, it follows at once that must be finite. Hence every open subset of G with compact closure is finite. In particular, every compact neighborhood of a point in G is finite, whence, since the topology of G is Hausdorff, we conclude that single points are open sets.

°

109

Therefore G is discrete, and part (i) of the theorem implies part (ii).

o

On the other hand, Ll(G) always contains what is kno~~ as an approximate identity, whether G is discrete or not. We give the definition of an approximate identity in the context of an arbitrary commutative Banach algebra.

net

Definition 4.7.2. Let A be a commutative Banach algebra. (u a ) C A is said to be an approximate identity if (i)

(ii)

sup lIu II < a a

go.

(x E A).

lim a lIu ax - xII = 0

Obviously, if

A

A has an identity

mate identity on setting

e,

then

(u) is an approxia The converse need not

u = e for all a. a be valid, as seen from the next theorem and Theorem 4.7.1.

in

Before we can prove the existence of an approximate identity we need one preliminary result. Ll (G)

Proposition 4.7.3. Let G be a locally compact Abelian topological group and let A be a Haar measure on G. If U is a nonempty open subset of G, then A(U) > o. I

Proof. Since A is translation invariant, we need consider only nonempty open subsets that contain the identity of G. Suppose U eGis such a set and A(U) = O. Then A(t + U) = 0, t ~ G. KeG is compact, then (t + U l t E K) forms an open covering of K, and so there exist t l ,t 2, ... ,t in K such that n n K C vk=l(t k + U). Consequently A(K) < ~=lA(tk + U) = O. Since K is arbitrary, it follows at once from the regularity of A that A = 0, which is a contradiction. If

Therefore

A(U) >

o.

o

4. The Gel'fand Representation of Specific Algebras

110

Theorem 4.7.2. group. Then LI(G)

Let G be a locally compact Abelian topological contains an approximate identity.

Proof. Let (u 1 denote the family of open neighborhoods of a o in G that have compact closure and are such that Ua = -ua . Such neighborhoods exist because G is locally compact. Then define denotes the characteristic ua = XujA.(U~, where as usual Xu function of U . In view of Proposftion 4.7.3, we see that each u ex a is well defined, and, as is easily verified, each u defines a a nonnegative element of Ll (G) n LCD(G) such that lIuaU I = 1. Moreover, (u) becomes a net provided we set a > e if and only if U C Uc • a a ~ 10 see that (ual is an approximate identity in Ll(G) we need only show that limJlua * f - fill = 0 for each fELl (G) • Let f E L1 (G). Then we see, on applying Fubini's Theorem [Ry, pp. 269 and 270], that lI

ua * f - fill

= IGIIG

f(t - s)ua(s) dA,(s) - f(t)1 dA,(t)

= IGIIG

[f(t - s) - f(t)]ua(s) dA,(s)1 dA,(t)

~IG[fG 'f(t - s) - f(t)luex(s) dA,(s)] dArt)

= IG[IG =

IG

11s(f)(t) - f(t)l dArt)] ua(s) dA,(s)

111s(f) - fIl1ua(s) dA,(s).

However, by Proposition 4.7.1(ii), 111s(f) - fill is a uniformly continuous function on G, whence, given e > 0, there exists some Uao such that, if s E Ucro' then 111s(f) - fli l < e. Consequently, if a > a, then o

lIua

* f - fill ~ <

Iu

a

sfu

111s (f) - fill ua(s) dA (5)

ua(s) dA(s) a = elluall l

III

= c.

Therefore {ua )

C

Ll(G)

is an approximate identity.

0

slight modification of the preceding argument using the fact that C (G) is norm dense in L (G), I < p < -, reveals that c plim Ilu * f - fll = 0, f E L (G), where (u 1 is defined as in the d a p p a proof of Theorem 4.7.2. The details are left to the reader. It should be noted that L (G), I < p < -, is a Banach algebra under p convolution only if G is compact. This is a nontrivial result [HR 2, pp. 469-472]. In any case, however, it is easily seen that supa lIua II p = -, 1 < P < -, when G is not discrete. This follows at once on noting that, if G is not discrete, then lim A(U ) = 0 11 a a' and lIu II = A(U 1- q where IIp + l/q = 1. a p a' A

Let us now turn our attention to describing the maximal ideal space of L1 (G). If T E 6{L I (G», then we know that T E LI(G)*, UTII ~ 1, and T(f * g) = T(f)T(g), f,g E L1(G). Since Ll (G)* may be identified with Lm(G) [HR I , p. 148; L, pp. 59 and 60], this implies the existence of a nonzero element y E L (G) such that T(f)

= IG

-

f(t)y(-t) dAft)

(f E Ll (G» •

Moreover, applying Fubini's Theorem [Ry, pp. 269 and 270], we see, on the one hand, that T(f * g)

= IG

f * g(t)y(-t) dAft)

= IG[jG f(t - s)g(s) dA(s)] y(-t) dA{t) =

IG

g(s)[jG f(t)y(-t - s) dA(t)] dA(s),

while, on the other hand, we have T{f * g) = T(f)T(g) =

IG

f(t)y(-t) dAft)

IG

g(s)y(-s) dA(s)

112

4. The Gel'fand Representation of Specific Algebras =

IG

(f, g E Ll (G)) •

g(s) [y{-s)iG f(t)y(-t) dArt)] dA(s)

From Proposition 4.7.2(i) we know that

IG

defines an element of C(G), Ll(G)

in

IG

L~(G)*

F 0,

T(f ) o

whence from the weak* denseness of

[L, p. 245] we conclude that, for each

f(t)y(-t - s) dArt)

for almost all

(s E G)

f{t)y(-t - s) dArt)

s E G.

= y(-S)!G

f E Ll(G),

f(t)y(-t) dArt)

In particular, if fo E LI(G)

is such that

we see that

JG fo(t)y(-t

- s) dArt)

y{ -s) = - - - - - - - - ,fG f 0 (t)y (-t) dA (t)

=

T[Ts{fo)] T{fo)

for almost all s ( G. Since the expression on the right-hand side of the preceding equations is evidently a continuous function of s, we may assume without loss of generality that the element corresponding to

T

E L (G) ~

is itself a continuous function, namely

= T[Ts{fo)]/T(fo )'

y(-s)

y

s E G.

But then, repeating the previous argument, we deduce from the identity

IG

f{t)y(-t - s) dArt)

= IG

f{t)y(-s)y(-t) dArt)

that y(-t - s) = y(-t)y(-s), t,s E G. Consequently for each T E ~(Ll(G)), there exists some y E C(G) yet

+ 5)

= y(t)y(s),

t,s E G, T(f)

= IG

~e

see that, such that

and f(t)y(-t) dArt)

Moreover, it is easily verified that

y

is unique.

(f €

L1 (G)).

113

Conversely, straightforward computations reveal that, if y E Cee) is such that yet + s) = y(t)y(s), t,s E G, then the formula T(f) defines an element zero.

= IG

f{t)y(-t) dA(t)

T E 6(L l (G))

provided y

is not identically

Before we summarize this discussion in the next theorem, we wish to look a little further at such functions y. So suppose y E C(G) is such that yet + s) = y(t)y(s), t,s E G. If Y is not identically zero, then yeO) = yeO + 0) = y(O)y(O) shows that yeO) = 1, and then 1 = yeO) = yet - t) = y{t)y(-t), t E G, reveals that yet) ~ 0 and y(-t) = l/y(t), t E G. Furthermore, we claim that ly{t)I = 1, t E G, on the following grounds: if there existed some t such that ly(t)1 > I, then ly(nt)I = ly(t)I n , n = 1,2,3, •.• , contradicts the boundedness of y. Hence ly{t)1 < 1, t E G. Combining this with the fact that y{-t) = l/y(t), t E G, we have ly(t)1

= 1,

t ~ G.

Thus we see that, if y E C(G) yet + s) = y{t)y(s), t,s E G, then (i)

(ii) (iii)

is not identically zero and

yeO) = 1. Iy(t) I = 1, t E G. y(-t) = l/y(t) = yet), t E G.

That is, each such y is a continuous homomorphism of e into r = {, I , E~, "I = 1). Because these homomorphisms of G play a central role in the study of Ll(e), as they can be identified with 6(L l (G)), we wish to give them a special name. Definition 4.7.3. Let G be a locally compact Abelian topolo• we denote the set of all continuous homogical group. Then by G • morphisms of G into r = {, , , E~, I" = 1). If y E G, we say that y is a continuous character of G.

4. The Gel'fand Representation of Specific Algebras

114

For notational reasons we wish to make • namely, if regarding the elements y E G; write ~(t) = (t,y), t E G. Note that with (t~y), t E G. The utility of this notation the sequel.

a special convention • we shall usually y E G, this notation, y(-t) = will become apparent in

Recalling the Gel'fand Representation Theorem (Theorem 3.3.1), we can summarize our development to this point in the following theorem: Theorem 4.7.3. group. (i)

Let G be a locally compact Abelian topological

The formula T(f)

= IG

f(t) (t,y) d~(t)

determines a bijective correspondence between A{LI(G))

•

•

and G.

•

(ii) In the Gel'fand topology on G, G is a locally compact • is a Hausdorff topological space. If G is discrete, then G compact Hausdorff topological space. (iii)

If f E LI(G),

the Gel'fand transform of f

•fey) = IG f(t) (t,y)

d~(t)

is given by

•

(y E G).

(iv) The Gel'fand transformation is a norm-decreasing homomor• of C (G) • that separates phism of LICG) onto a subalgebra LI(G) • o. the points of G. If G is discrete, then LI(G) contains the constant functions. As an immediate corollary we have the next result. Corollary 4.7.1 (Riemann-Lebesgue Lemma). Let G be a locally compact Abelian topological group. If f (LICG), then • limy_ fey) = 0; that is, given ,> 0, there exists some compact set KeG such that If(y)1 < e, y E G - K.

..

-

.

lIS Because of the identifications indicated in Theorem 4.7.3 we " as the maximal ideal space of L1 (G) shall generally speak of G " and of the Gel'fand transform of f E LlCG) as a function on G. Note also that the norm-decreasing nature of the Gel'fand trans-

" IIfIlCD:5 IIflll, f E Ll(G).

formation means explicitly that

For the classical groups, for example, r and ~ the Gel'fand transform on LICG) is precisely the familiar Fourier transform. " For this We shall see this momentarily when we describe r" and ~ reason we shall use the terms "Gel'fand transform" and "Fourier transform" interchangeably in discussing Ll!G). Also, recalling Example 1.2.9, it is natural to write LICG) = FLl(G). Before we proceed any further with the general investigation " "r. Suppose y E ~ " of LICG), let us describe in detail Rand Since (O,y) = 1, there exists some 6 > 0 such that

J~ (s,y) ds ~ O. Consequently we deduce from

~ (t

+

s,y) ds = (t,y)~ (s,y) ds

(t E IR)

that

Ct,y)

=

s~ (t

=

y' (t)

=

s,y) ds

Jo (s,y)

Jt+6 t Thus we see at once that y

+

6

ds

(s,y) ds

J6o Cs,y)

(t E lR) •

ds

is differentiable on

lim (t h-O

+ h,y) -

h

(t,y)

~

and so

116

4. The Gel'fand Representation of Specific Algebras

= (t,V)

lim (h,V) ~ (O,y)

h-O

= V' CO)Vet)

(t (lR).

Solving this elementary differential equation and using the facts vCO) = I and IVCt) I = I, t (!h, we see that there exists some ~ E lR such that y (t) = (t, V) = ei~t, t E iRe Conversely, it is obvious that for each ~ E 1f\. the function ei~. belongs to R, so that ~ = {e igo I ~ E u~. Since the correspondence ~ - e ig ., g E ~ that

" with the point set ~, and is bijective, we can also identify lit we shall often do this. It is now evident that the Gel'fand transform of

f E

Llor~

is indeed the classical Fourier transform .

..

The description of f

is obtained most easily by utilizing the

preceding results. To do this we note that r is isomorphic to iV2nZ, where 2n£ = (2nk I k ~ Z). Then it is easy to check that .. " V ( f if and only if V E~ and V is periodic with period 2n. Thus we see that = (e ik . IkE if). As in the case of lri, we

r ..

generally identify f k E oZ.

with L

.

by means of the mapplng

k - e

ike

,

..

" If V E Z, To round out this discussion we shall describe L. then there is some f such that (1,V) = ,. Clearly then we must

,E

have (k,V) = ,k, k E L. Conversely, given ,E r, the mapping k - ,k, k E Z, is obviously an element of Thus we see that 1 = (,Co) I , E fl. As usual, we generally identify ~" with f.

Z.

.."..

Allowing for a certain amount of imprecision, we observe that ..,," "" (JR.) = l[( = lR, (f) = It. = f, and (~) = f = lL.. These observations are indicative of a general phenomenon for locally compact Abelian " topological groups G, which, loosely speaking, says that (G) = G. We shall return to this important result, known as the Pontryagin Duality Theore~ in Section 10.5, where we shall discuss it in detail.

..

..

It is obvious that for the algebras previously discussed in

117 this chapter the Gel'fand transformation is injective, that is, the algebras are semisimple. The algebra L (G) is also semisimple, 1

but the proof of this fact is not entirely trivial. Theorem 4.7.4. group.

Then

LI(G)

Let

G be a locally compact Abelian topological

is semisimple.

Proof. In view of Theorem 3.5.1 and Corollary 3.5.1, it suffices to show that, if f E L1 (G) is such that limnllfnll~/n = 0, then f = 0, where by fn we mean the n-fold convolution of f with itself. We claim first that we may assume, without loss of generality, that f E LI(G) n Lm(~). Indeed, suppose we know that f E LI(G) n LmCG) and limnllfDlI~/n = 0 imply f = O. Then suppose g E LI (G) is such that lim IIgnlll1/n = O. Let {u 1 C LI (G) n L CG) be an approximate n a m identity; for example, we may choose {u) as in the proof of a Theorem 4.7.2. From Theorem 3.5.1 we see that, for each a, limnliCua * g)nll~/n = 0, as Rad[L I (G)] is an ideal. However, by the assumption just made and the fact that lIu a * gil 0 )<- lIu aO) II IIgll1 for each 0, we deduce that u * g = 0 for each o. But a lim Uu * g - gill = 0, whence g = O. Thus to prpve the theorem we o a 1 need only show that f E LICG) n Lm(G) and limnllfnll~ n = 0 imply f

= o.

To prove this we note first that, since f E LICG) n Lm(G), we have f E L2 CG). Furthermore, setting f*(t) = f(-t), we see that f* E Ll(G) n Lm(G) whenever f E Ll(G) n Lm(G). From Example 1.2.7 and Proposition 4.7.2 we see that, if f E LICG) n Lm(G) , then f * f* E LI(G) n LmCG), and f * f* * g E C(G) for each g E LI (G). Consequently it is evident that F(g) = f * f* * g(O), g E LI(G), defines a linear functional F on Ll(G), and

Next we define, : LI(G) x LI(G) - ~ by ,(g,h) = F(g * h*), g,h ~ LI(G). We claim that , is a nonnegative symmetric bilinear form on Ll(G).

4. The Gel'fand Representation of Specific Algebras

118

Indeed, if g E LI(G) , ,(g,g)

then ~

F(g * g*)

=f

*

f* * g

*

g*(O)

= (f *

g)

= (f *

g) * (f

= Ie

f

*

= IG

If * g(_t)1 2 dA(t)

* Cf* *

g(-t)f

g*)(O)

* g)*{O) *

g(-t)

dAft)

> o. Thus V is nonnegative.

If g,h E tICG),

then

yCg,h)

= F(g *

h*)

= f • f*

= (f

iG = IG

=

* g * *

h*(O)

* h)*(O)

*

g)

f

* g(-t)f * h(-t) dA(~)

(f

f * g(-t)f * h(-t) dAft)

= .(h,g). Hence •

is symmetric.

The bilinearity of • is easily established, and we conclude that • is a nonnegative symmetric bilinear form on LICG). Consequently the Cauchy-Schwarz Inequality is valid for , [L, p. 373]; that is, IF(g * h*)1 2

~ F{g * g*)F(h • h*)

119

Now let [ual c L1 (G) n L.(G) be the approximate identity constructed in Theorem 4.7.2. Then u*; U aDd a a F(ua * u*) a'

for each

=f *

f*

* ua * u*(O) a

<

IIf * ffrU.Uua *

<

Hf * f*ii.

u~1l1

as Uualll = 1. Combining this observation with the of F and the Cauchy-Schwarz Inequality, we deduce that

at

~ontinuity

Setting K = (llf • f*li.)1/2 and applying this inequality to PCg if g*) ~ 0, we see that IF(g)1 < K[F(g * g*)1/2]

~ K1+ l / 2 [F(g * g* * g * g*)1 / 4 J Continuing in this fashion, we find that

IF(g)l Sxl+l/2+".+1/2n[F([g * g*J 2n)1/2n+l]

:s Kl+1 I 2+ ... +1 / 2n [lif * f*llCDIl (g

for each g E L1 (G)

,Feg)I

and n

= 0,1,2, .•••

< 1C3 [limll{g n

*

In particular, if g = f

*

Thus we have

n 1/2n g*)2 ]1/2

III

*

f*,

Fef • f*)

=f *

(g E L1 eG).

then g * g* E Rad[L1 (G)J,

the radical is an ideal, and we conclude that

o=

n / n+l g*)2 Ill) 1 2

f* * f * f*(O)

as

4. The GeI'fand Representation of Specific Algebras

120

However, since f E L1 (G) n Le(G) , we see from Proposition 4.7.2(i) that f * f* E C(G), and so the preceding identity implies that f * f*(t) = 0, t E G.

o =f

Hence

from which we deduce that f =

= IG

* f*(O)

fCt)

If(t)12 dA(t) ,

=0

for almost all

t E G;

that is,

o. Therefore

LICG)

is semisimple.

Some rather easy corollaries of this theorem are worth mentioning explicitly. The first is just a rephrasing of the theorem.

..

.

Corollary 4.7.2. Let G be a locally compact Abelian topological group. If f E LI(G) and fCY) = 0, y E G, then f = o. Corollary 4.7.3. Let G be a locally compact Abelian topological group. Then G separates the points of G.

..

.

Proof. Suppose G does not separate the points of G; that is, suppose there exist t,s (G, t ~ s, such that (t,y) = (s,y), y E G. Clearly then (t - s,Y) = 1, y E G.

.

.

But then for any f E LI(G)

..

[Tt_s(f)] (y)

we would have

= fG

Tt_s(f)(u) (u,y) dA(u)

= IG

f[u - (t - s)](u,y) dA(u)

= IG

f(u)(u

= (t

s,y)

.

= f (y)

+

t - s,y) dA(u)

JG feu) (u,Y)

dA(u)

.

(y E G).

121 Since LI(G) is semisimple, this would imply that Tt_s(f) fELl CG), which is obviously false if t ~ s.

= f,

A

Thus G separates the points of G.

o

A

The set of continuous characters G can also be given a group A structure under which it is an Abelian group. Namely, if y,w E G, one defines (t,y + w) = (t,y)(t,w) and (t,-y) = (-t,y), t ( G. A It is easily seen that with these operations G does indeed become an Abelian group. The details are left to the reader. The identity A element of the group 0 is the continuous character Yo such that y (t) = 1, t E G. We shall always use y to denote this element o A 0 of G. A

Considered as the maximal ideal space of LI(G), G is endowed with a topology -- namely, the Gel'fand topology -- and a natural A question to ask is whether the Gel'fand topology on G and the group A A structure of G combine to make G a topological group. That this is indeed the case will be seen in the corollary to Theorem 4.7.5. A First, however, we wish to use the group structure of G to derive two further corollaries of Theorem 4.7.4. Corollary 4.7.4. Let G be a compact Abelian topological group and suppose Haar measure A on G is normalized so that A(G) = 1. A Then G is a complete orthonormal set in the Hilbert space L2 CG). A

Proof.

Clearly G C L2 (G) , A If y = Yo E a, then

IG

(t,y) dAft)

A

as G C C(G)

and G is compact.

= ACG) = 1,

A

whereas, if yEO and y ~ Yo' then there exists some s E G such that (s,y) ~ 1. Hence, since A is translation invariant, we see that

IG

(t,y) dAft)

= IG =

(t

+

s,y) dAft)

(S,y)JG (t,y) dA(t),

122

4. The Gel'fand Representation of Specific Algebras

which can clearly be valid only if

• Consequently, if y,w E G,

IG

Thus

we see that

(t,y)(t,w) dA(t)

•

G is an orthonormal set in

To show that f E L2 (G) and

a•

= IG

(t,y - w) dA(t)

= I

for

= 0

for y

y = ~

W

w•

L2 (G) .

is complete it suffices to prove that, if

fG f(t) (t,y) dA(t)

" (y E G),

= 0

then f = 0 [L, p. 398]. But the left-hand side of this equation • is just fey) since L2 (G) C LI(G), as G is compact. The desired conclusion now follows immediately from the semisimplicity of LI(G).O In general, when G is a compact Abelian topological group, we shall always assume that Haar measure A is normalized so that A(G) = 1. Cerellary 4.7.5. Let G be a locally compact Abelian topolo• is compact, and if G is gical group. If G is discrete, then G " is discrete. compact, then G Proof. If G is discrete, then, by Theorem 4.1.1, has • an identity, whence a is compact by Theorem 3.2.2. On the other • hand, if G is compact, then G C LI(G). Hence, by Corollary 4.1.4, " we deduce that, for each w E G,

" w(y)

= IG

(t,w)(t,y) dA(t)

=I

for y

= 0 for

=w

y ~ w

" (y E ct).

123

• Thus the characteristic function of each singleton set (w), w E G, • and so (w) is an open set for is a continuous finction on G, • each w E G.

• is discrete. Therefore G

o

The converse of the results of Corollary 4.1.5 is also valid, and we shall come back to this point in Sections 1.3 and 10.5. Now let us return to the question raised before Corollary 4.1.4: • Is the algebraic group G a topological group in the Gel'fand topology? Since the Geltfand topology is Hausdorff, we need only show that the group operations are continuous in the Gel'fand topology in order to obtain an affirmative answer. To accomplish this we shall first describe an alternative description of the Gel'fand topology • which is of considerable interest in itself. The fact that on G • with the Gel'fand topology is a locally compact Abelian topologiG cal group will be an easy consequence of this description. Theorem 4.7.5. Let G be a locally compact Abelian topological group. For each a > 0 and each compact set KeG define U(K,a)

= (y

•

lyE G, I(t,y) -

11

< a, t E K).

Then the family of sets (U(K,e)), where e > 0 is arbitrary and KeG is an arbitrary compact set, forms a neighborhood base at • for the Gel'fand topology on G. • y EG o

Proof. It suffices to show that every such set U(K,a) is • open in the Gel'fand topology on G and that every open neighborhood of y in the Gel'fand topology contains some such set. o

• - f, defined by We first observe that the function from G X G • (t,y) - (t,y) = yet), t E G, yEO, is continuous. Here (t,y) • and G X G • is given denotes a point in the product space G X G, • the product topology. Indeed, suppose (t) C G and (~) C G ~ a are nets that converge to t (G and y E OJ respectively. If

4. The Gel'fand Representation of Specific Algebras

124

A

f (Ll(G)

is such that

exists some

eto

fCy)

a >

such that

A

then, since

0,

~

implies

eto

A

f ( C (0),

A

0

f(y~ ~

O.

there

Elementary

computations reveal that for each s E G we have A

= [T-s (f)]

(s,y)f(w)

A

A

(w)

(w

E G),

whence, recalling the development preceding Theorem 4.7.3, we conclude that A

(s,y)

=

[T -s (f)] (y)

(s ( G)

A

fey) and A

(s'YJ

=

[T -s (f)] -(Y~

(s E G; a > a ).

•

0

f(YJ

Clearly then to show that lim (t ,y ) = (t,y) we need only prove • a a.a • • that lima[T_ta(f)] (Ya) = [T_t(f)] (y) since limaf(Ya) = fey). But

+

I [T_t(f)]

•

A

(Ya) - [T_t(f)] (y)1

< II T_t (f) - T_ t (f)1I 1 a

from which we at once deduce the desired conclusion via Proposition • to r is 4.7.I(ii). Thus the mapping (t,y) ~ (t,y) from G X G continuous. Now suppose s > 0 and KeG is compact, and consider U(K,s). If w E U(K,s), then from the continuity of the function (t,y) ~ (t,y) t E G,y E G, and the fact that I(t,w) - 11 < s, t E K, we deduce that, for each t E K, there exists an open neighborhood Vt of t A

125

in G and an open neighborhood

W

u"

,.

of w in G such that, if

t

(s,Y) ~ Vt x Ww,t' then I(s,y) - 11 < i . The sets {V t I t E K) clearly form an open covering of the compact set K, and so there t l ,t 2, ..• ,tn in

exist

K C ~=IVtk'

K for which

Let

W= nk=IWw,tk.

Evidently W is an open neighborhood of w in the Gel'fand topology, and if t ~ K, then I(t~y) - 11 < i, y (W; that is, We U(K,i). Since ~ E U(K,s) is arbitrary, we conclude that U(K,e) is open in the Gel'fand topology. the other hand, suppose that W is an open neighborhood of Yo in the Gel'fand topology. Then~ since the GeI'fand topology is the relative weak* topology on A(L 1 (G)), we may assume without loss of generality that there exist f l ,f2 , ••• ,fn in LI(G) and £ > 0 such that ,.,. ,. ,~= {y I y.E. G, Ifk(y) - fk(yo)1 < e, k = 1,2" •.• ,n). On

Let gk ~ ec(G) and consider

If yEW. o

be chosen so that

llfk - gklll < e/3, k = l,2" ...• n,

then

+

~

II +

,. ,. Igk(y o) - fk(y o ) I

ll

,.

,.

f k - gk l + Igk(Y) - gk(Y o) I

ligk

- fklll

e e e 333

<-+-+-

=£ whence W c o

w.

(k = 1,2, ... ,n),

126

4. The Gel'fand Representation of Specific Algebras

Now let KeG be any compact set such that gk(t) = 0, t t K, k = 1,2, ... ,n. For example, K can be taken to be the union of the " we have supports of gk' k = 1,2, ..• ,n. Then for each y € G 19k(y) - gk(YO) 1 = lJG gk(t) (t,y) dAft) -

SG

gk(t) (t,yo) dA(t)l

~IK 19k(t)ll(t,y) - 11 dA(t)

< sup l(t,y) - llllgklli tEK Consequently, if e' = e/(6max k=I,2, ... ,n!lgklll) then

(k

= 1,2,. .. ,n).

and y E U(K,e'),

and so U(K, e') ewe '\'. o

Therefore the sets U(K,e), £ > 0 and KeG compact, form a neighborhood base at Yo for the Gel'fand topology. 0

" A moment's reflection reveals that the Gel'fand topology on G is that topology in which convergence of a net of continuous characters of G is equivalent to the convergence of the net of continuous characters, qua functions, on compact subsets of G. It is often " in this way. useful to think of the Gel'fand topology on G The promised result about

" can now be proved. G

Corollary 4.7.6. Let G be a locally compact Abelian topolo" with the Gel'fand topology and the group gical group. Then G operations (t,y

+ ~)

Ct,-y)

= (t,y)(t,w), = (-t,y)

is a locally compact Abelian topological group.

(t

" E G; y,w € G),

4.8. M(G)

127

•

Proof. Since G is a locally compact Hausdorff topological space in the Gel'fand topology, we need only show that the group operations are continuous with respect to this topology. This is easily verified by using the description of the Gel'fand topology provided by Theorem 4.7.5 and observing that I(t,y

+

w) -

11

< I(t,y

+

w) - (t,y)1

=

I (t ,y) (t,w)

=

ICt,w) -

11

(t,y)1 +

I(t,y) - 11

+

I(t,y) -

+

I(t,y) - 11

11

(t E G;

• y,w E G)

and that I(t,-y) - 11 = I(t,y) =

11

•

I(t,y) - 11

(t E G; yEO).

The details are left to the reader.

[j

• is considered as a locally compact Abelian topological When G group, we shall call it the dual group of G.

..

..

Following Theorem 4.7.3 we observed that ~= ~ r = Z, and L = f. The reader should verify that the Gel'fand topology on ~ • and L• coincides with the usual topology on F~ Z, and r, f, respectively. Thus as dual groups of ~ r, and Z we see that lR = IF\, r = Z, and Z = r.

-

...

.

.

4.8. M(G). As indicated in Example 1.2.8, the space M{G) of bounded regular complex-valued Borel measures on a locally compact Abelian topological group G is a commutative Banach algebra with identity. The algebra operation is, of course, convolution of measures. Consequently, by Theorem 3.2.2, ~(M(G)) is a compact • then it is easily verified Hausdorff topological space. If y E G, that

128

4. The Gel'fand Representation of Specific Algebras (~

defines an element T E aCMCG)), a subset of a (M (G) ) •

E M(G))

" can be identified with and so G

• = a(M(G)), In the case that G is discrete it is evident that C " is a proper as M(G) = LI(G). But if G is not discrete, then G subset of a(M(G)) and, moreover, ~" is not dense in a(M(G)). We shall not prove these results, but cite them only to illustrate that the structure of the maximal ideal space can be more complicated than the discussion of the preceding sections might indicate. For some further discussions of these results and the maximal ideal space of M(G) the reader is referred to [DkRa, pp. 1-28; GRk~, pp. 176-192; H, p. 143; HR l , pp. 368 and 369; Ri, pp. 328-330]. Nevertheless, M(G) is still a semisimple algebra, and indeed it can be shown, as we shall do in Section 10.5, that, if ~ E M{G) and

" ~(y)

= IG

(t,y) d~(t) = 0

" (y E G),

" defined here is casily seen then ~ = O. The function ~" on G to be a bounded uniformly continuous function on 0" and it is generally called the Fourier-Stieltjes transform of ~.

CHAPTER 5 SEMISIMPLE COMMUTATIVE BANACH ALGEBRAS 5.0. Introduction. In this chapter we shall concentrate our attention mainly on various semisimple commutative Banach algebras. First we shall obtain some results concerning the case in which the Gel'fand transformation of a semisimple commutative Banach algebra is either a topological isomorphism or an iso~etry. The chief tool in obtaining these results will be a theorem asserting that the spectral radius IIxll a = IIxll CD defines an equivalent norm on a semisimple commutative Banach algebra A if and only if IIxll 2 < Kllx 211, x E A, for some constant K > O.

..

In Section 5.2 we shall see how to make the Gel'fand representation A of a semisimple commutative Banach algebra A into a Banach algebra that is isometrically isomorphic to A and has the same maximal ideal space as A.

..

Following this we shall discuss several results about homomorphisms and isomorphisms between commutative Banach algebras. The most interesting of these results asserts that an algebra homomorphism of a commutative Banach algebra into a semisimple commutative Banach algebra is automatically ~ontinuous. An easy corollary of this theorem will show that the norm in a semisimple commutative Banach algebra is essentially unique. Finally, we shall employ some of these results, together with our knowledge of topological zero divisors garnered in Section 1.6, to show that in a self-adjoint semisimple commutative Banach algebra with identity an element is singular if and only if it is a topological zero divisor. 129

s.

130

Semisimple Commutative Banach Algebras

5.1. The Gel'fand Representation of Semisimple Commutative Banach Algebras. From the definition of semisimplicity we know that the Gel'fand transformation of a semisimple commutative Banach algebra A is injective, and so the Gel'fand representation is a subalgebra of C [aCA)] that is isomorphic to A. Furthermore, by the o Beurling-Gel'fand Theorem (Corollary 3.4.1), we see that, if A is a semisimple commutative Banach algebra, then the spectral radius IIxllo = IIxll ~ of x in A defines a norm 11·11 a on A, called the spectral radius norm. It is evident that A is a normed algebra with the spectral radius norm and that IIxll 0 <- IIxli, x E A. However, (A,II·lto ) need not be a complete normed linear space; that is, (A,II·lI a ) need not be a Banach algebra. Obviously, if 11.11 0 and 11·11 were equivalent norms on A (that is, if there existed some

..

K > 0 such that IIxll ~ Kllxllo' x E A), then {A, II· "0 ) would also be a Banach algebra, and our first theorem provides a necessary and sufficient condition for this. We shall then use this result to derive conditions under which the Gel'fand transformation is a topological isomorphism. Similar arguments will give us conditions under which the Gel'fand transformation is an isometry. Theorem 5.1.1. Let A be a semisimple commutative Banach algebra. Then the following are equivalent: (i) (ii)

x

~

11·11

and

11.110

are equivalent norms on A.

There exists some constant

K > 0 such that

IIxll 2 < Kllx 211,

A.

Proof. exists some

If 11·11 and 11.11 0 are equivalent norms, then there Kl > 0 such that IIxli < Klllxllo' x E A. Consequently (x ~

setting part (ii). On

we see that part (i) of the theorem implies

Conversely, suppose

lIxU 2 ~ Kllx 2U, x € A,

for some

K>

o.

A).

5.1. The Ge1'fand Representation

131

Then applying this estimate repeatedly, we see that, for each x E A, IIxll ~ KI/2I1x 2111/2

< Kl/2[Kl/2UX4111/2] 1/2 =

222 Kl/2+l/2 IIx 2 111/2

< •.. < Kl / 2+1/22 +... +1/ 2n ll x2nlll/2n

-

-

Cn = 1,2,3, ••• ).

Thus, by the Beurling-Gel'fand Theorem (Corollary 3.4.1) or the Spectral Radius Formula (Theorem 2.2.2), we deduce that IIxll < KUxll , a x E A. This estimate, combined with the fact that IIxU a < IIxll, x E A, shows that 11·11 and 1I'lIa are equivalent norms. Hence part (ii) implies part (i).

o

The first application of this theorem is the next corollary. Recall that a mapping is said to be a topological isomorphism if it is both an isomorphism and a homeomorphism. Corollary 5.1.1. Let A be a semisimple commutative Banach algebra. Then the following are equivalent:

• of A is a closed subal(i) The Gel'fand representation A gebra of (Co[fl(A)],U·Um). (ii)

There exists a constant

K > 0 such that

IIxU 2 ~ Kllx 2U.

x E A. (iii) The Gel'fand transformation is a topological isomorphism • of A onto A. Proof. If part (i) holds, then the Gel'fand transformation is evidently a continuous bijective linear mapping of A onto the Banach space •A C C [fl(A)]. Thus an application of the Open Mapping o Theorem [L, p. 187] reveals that the inverse of the Gel'fand trans• to A, and so formation is a continuous linear mapping from A • there exists some Kl > 0 for which IIxll ~ K111xllm = K1llxlla. x E A.

S. Semisimple Commutative Banach Algebras

132

Consequently 11·11 and 11.11 0 are equivalent norms on A, whence, by Theorem 5.1.1, we conclude that there exists some K > 0 such that IIxll 2 < Kllx 2l1, x 'A. Thus part (i) implies part (ii). If part (ii) holds, then Theorem 5.1.1 shows that 11·11 and 11.110 are equivalent norms, from which it follows immediately that the Gel'fand transformation is a topological isomorphism, as

•

IIxlia = lIxllm, x E A. (iii).

Hence part (ii) of the corollary implies part

To show that part (iii) implies part (i) is even easier and is left to the reader.

o

Before we can state the next corollary we need a definition. Definition 5.1.1. Let A be a commutative Banach algebra. •• A is said to be self-adjoint if whenever • x E• A then x E A, where the bar denotes complex conjugation. Thus A is self-adjoint if and only if the Gel'fand representa• tion A is closed under complex conjugation. It is left as an exercise for the reader to verify that the Banach algebras C (X), X o being a locally compact Hausdorff topological space, and Ll(G), G being a locally compact Abelian topological group, are self-adjoint, whereas A(D) is not. It should be observed tha~ if A is a self-aajoint commutative Banach algebra, then an appeal to the Stone-'Veierstrass Theorem • [L, p. 332] shows that the Gel'fand representation A is dense in

•

•

(Co [~(A)] , I;· ltD). In particular, Ll (G) is dense in Co (G) , G is a locally compact Abelian topological group.

where

The proof of the next corollary utilizes the general fact about self-adjoint algebras just mentioned. The details are left to the reader.

..

S.2. A as a Banach Algebra

133

Corollary S.1.2. Let A be a self-adjoint semisimple commutative Banach algebra. Then the following are equivalent: (i) x € A.

Cii)

K > 0 such that

There exists a constant

..

A

Ilxll 2 < Kllx 211,

= Co [deAl].

The final result of this section gives a necessary and sufficient condition for the Gel'fand transformation to be an isometry. Theorem S.1.2. Let A be a commutative Banach algebra. the following are equivalent:

(ii)

Then

-

The Gel'fand transformation is an isometry of A onto A.

Proof. If IIxU 2 = IIx 2U, x E A, then clearly for each x E A I 2n II = IIxll 2n , n = 1,2~.3, .... Hence, by the 8eurlingwe would have Ix Gel'fand Theorem (Corollary 3.4.1), we deduce that n

IIxll

= limUx 2

II

1/2n

n

.. = IIxll

(x E A),

G)

from which it is apparent that the Gel'fand transformation is an isometry. The converse is immediate on noting that 2

IIx II

-2

= IIx

II CD

.. 2

= IIxll = IIxll

2

(x E A)

CD

whenever the Gel'fand transformation is an isometry.

c

It is worth remarking that, if the Gel'fand transformation is an isometry, then A must be semisimple and A must be a closed subalgebra of C [A(A)).

-

..

o

A ~~ Banach Algebra. If A is a commutative Banach algebra, then the Gel'fand Representation Theorem (Theorem 3.3.1) 5.2.

s.

134

Semisimple Commutative Banach Algebras

• is a normed subalgebra of C [A(A)]. It need not, shows that A o however. be a Banach algebra with ~he supremum norm. Nevertheless, • can be normed in such a way that it does if A is semisimple, A become a Banach algebra. This technique is often useful in studying various problems. Theorem 5.2.1.

• bra and define IIxli. (iJ

(ii) (iii)

11·11.

••

= IIxll, x E A.

Then

•

is a norm on A.

• (A,II·II.) • A(A)

Let A be a semisimple commutative Banach alge-

is a semisimple commutative Banach algebra .

is homeomorphic to

A(A).

(iv) The Gel'fand transformation is an isometric isomorphism of • A onto (A,II·U.).

• and that The verification that is a norm on A (A,II·II.) is a commutative Banach algebra is routine and is left to the reader. The semisimplicity of A is required to show that II·U. is well defined. Proof.

n·lI.

•

•

•. If T E A(A), define wT on A by w (x) = T(X), •x E A •T • Clearly wT is a complex homomorphism of A; that is, wTEA(A). • Moreover, the mapping from A(A) to A(A) so defined is bijective. • • •• Indeed, if wT = W , then X(T I ) = TI(X) = T 2 (X) = X(T 2), x E A, 1 T2 • which implies that TI = T2 , as A separates the points of A(A) (Theorem 3.3.1). Thus the mapping is injective.

• and one defines T(X) = w(x), • w E A(A) • • • • x E A, then obviously T E A(A) and wT(x) = T(X) = w(x), x E A. Hence the mapping is surjective. On the other hand, if

Furthermore. if (T J C A(A) is a net that converges to T E A(A), • a • then, since A C C [A(A)], we see at once that {w (x)] converges Ta to wT(x) for each x E A. from which it follo~s by the definition • that (w ] converges to w • of the Gel'fand topology on A(A) T T a

.0. .

5.3. Homomorphisms and Isomorphisms

135

.

Consequently the mapping T ~ wT of 6(A) onto 6(A) is continuous. A similar argument proves the continuity of the inverse mapping . Therefore 6(A) and 6(A) are homeomorphic •

..

The semisimplicity of are now apparent.

..

(A,II·II .. )

and part Civ) of the theorem

o

One of the advantages of this construction is that it enables us to replace a semisimple commutative Banach algebra A by another such algebra B, where the algebra B is an algebra of continuous functions under pointwise operations that is isometrically isomorphic to A and has the same maximal ideal space as A. The algebra B, as we have just seen, is B = A with the norm 11·11.. Thus, for example, if A = Ll(G), G being a locally compact Abelian topological group, then LI(G) is isometrically isomorphic to the semisimple commutative Banach algebra L1CG) of functions in C CG) with the .. 0 norm IIfll .. = IIf1l1, f (; LI(G). A special case of this was mentioned in Example 1.2.9 when G = f and G = Z. There we denoted LICf) by FLICf). Similarly in this case we may write LICG) = FL1CG) when G is an arbitrary locally compact Abelian topological group.

..

..

..

..

..

..

5.3. Homomorphisms and Isomorphisms of Commutative Banach Algebras. In this section we wish to prove several results concerning homomorphisms and isomorphisms between commutative Banach algebras. A corollary of the first of these results, which asserts that an algebra homomorphism of any commutative Banach algebra into a semisimple commutative Banach algebra is continuous, will show that the norm on a semisimple commutative Banach algebra is essentially unique. This assertion will be made precise in Corollary 5.3.1. First, however, we prove the indicated theorem. Theorem 5.3.1. Let A and B be commutative Banach algebras and suppose A is semisimple. If T : B ~ A is an algebra homomorphism, then T is continuous.

S. Semisimple Commutative Banach Algebras

136 Proof.

Since

T is a linear mapping

and

A and

Bare

Banach spaces, we may appeal to the Closed Graph Theorem [L, p. 189] to deduce the continuity of T. To this end, suppose {xk ) is a sequence in B and x E Band yEA are such that

and lim/lT(X k) - yllA k

where II'II B and We must show that

= 0,

U'IIA

denote the norms in T(x) = y.

B and

A,

respectively.

Let T E ~(A) and set wT = ToT. Since T is a homomorphism, it is easily seen that either wT = 0 or wT E ~(B). In either case we have limkw,,(x k) = wT(x). lienee we see, on the one hand, that for each T E A(A) =

T(x) " (T)

and, on the other hand,

" lim T(xk ) " (T) = yeT), k

as simplicity of A,

" whence, from the semiT(x) " = y, we conclude that T(x) = y. Thus

Therefore, T is a closed linear mapping and hence, by the Closed Graph Theorem [L, p. 189], a continuous mapping. Corollary 5.3.1.

Let

c

A be a commutative Banach algebra under

the norms 11'11 1 and 11'11 2 . If (A,II·II I ) and 1i'11 2 are equivalent norms on A.

is semisimple, then

11'11 1

Proof. Consider the mapping T : (A,II'1I 2) - (A,li'H I ), defined by T(x] = x, x E A. F.vidently T is an algebra homomorphism and, by Theorem 5.3.1, is continuous. Thus there exists some K > 0 such

5.3. Homomorphisms and Isomorphisms

137

that IIxlll < Kllxll2, x ~ A. However, since (A,II·lI l ) and (A, 11.11 2) are both Banach spaces, this implies, via the Two Norm Theorem [L, p. 190], that

11.11 1 and

11.11 2 are equivalent.

0

Thus the meaning of our previous remark concerning the essential uniqueness of the norm in a semisimple commutative Banach algebra should now be clear. Corollary 5.3.1 is actually valid even if the assumption of commutativity is dropped, but the proof requires more of the general theory of Banach algebras than we have at our disposal. A proof of this result can be found in [Jo]. In Theorem 4.1.4 we proved that two compact Hausderff topological spaces X and Yare homeomorphic if and only if C(X) and C(Y) are algebraically isomorphic. Our next theorem shows that one~half of this equivalence is valid in general for commutative Banach algebras. Theorem 5.3.2.

Let

A and

B be eommutative Banach algebras.

If there exists an algebra isomorphism of is homeomorphic to 6(B). Proof. Suppose As before, for each

B onto

A,

then

6(A)

T: B ~ A is a surjective algebra isomorphism. T E 6(A) we define wT E 6(B) by

WT(X) = (T

0

T)(x) = T[T(x)]

(x E B).

Note that wT ~ 0, because if wT(x) = T(T(x)] = 0, x E B, then T = 0, as T is surjective, contradicting the fact that T E 6(A). Evidently the mapping ~(T) = wT ' T E 6(A), maps 6 (A) into 6(B). r.toreover, ~ is bijective. Indeed, if ~(Tl) = ~(T2)' then Tl[T(x)] = T2 [T(x)], x E B, whence we conclude that Tl = T2 , as T is surjective; and if w E 6(B), then, as above, we see that T (y) = W[T-l(y)], yEA, W defines an element TUI E A(A). Clearly ~(T) = w, and so ~ is W bijective.

138

5. Semisimple Commutative Banach Algebras

Furthermore, ~ and ~ -1 are continuous. For instance, suppose (T ) C A(A) is a net that converges to T E A(A). Then, since • a A C Co[A(A)], we see that for each x E A lim a

~(T~(X)

a

a

a

whence we conclude that ~ is continuous. tinuous. Therefore

•

•

= lim T [T(x)] = lim T(x) (T ) = T(x) o(T) = ~(T)(X)

~:

A(A)

~

A(B)

~

Similarly ~-l

is a homeomorphism.

is con-

o

A few remarks about this theorem are in order. First, if A is semisimple, then, by Theorem 5.3.1, T is continuous, and T is completely determined by the equation T(K) • (T) = •X[~(T)] where T E A(A) and x E B, and ~ is the homeomorphism defined in the proof of the theorem. Thus, if Tl and T2 were surjective algebra isomorphisms from B to A which defined the same homeomorphism ~J then Tl = T2 " This follows at once from the semisimplicity of •• • A and the equations Tl(x) (TJ = X[~(T)] = T2 (x) (T), T E A(A) and x E B. Contrary to the situation for C(X) discussed earlier, the converse of Theorem 5.3.2 need not be valid. In particular, if ~ : A(A) ~ ACB) is a homeomorphism between A(A) and A(B), then • T : B ~ A, defined by T(x) • (T) = X[~(T)], T E A(A) and x E B, need not define a surjective algebra isomorphism. The crux of the difficulty is that, given a homeomorphism ~: A(A) - A(B), it is not at all evident that x• 0 ~ (A• when • x E•B. The question of precisely which homeoMorphisms between A(A) and ACB) induce isomorphisms between B and A is generally a rather intricate one, even in the case that A = B. For some specific algebras the answers are known, for instance, when A = C(X) and B = Cry), X and Y being compact Hausdorff topological spaces. We shall not discuss the problem any further, but content ourselves ~ith describing the situation for LIOR) " Recall that A(LIOR» = ~

5.4. Characterization of Singular Elements Theorem 5.3.3. If T is a mapping from then the following are equivalent: (iJ

T

(ii)

~

Ll (lH) - Ll (lR)

139 LIOR)

to itseff,

is a surjective algebra isomorphism.

R-R is a homeomorphism of R onto itself of the form C9(t) = at + b, t E IR, for some a,b E IR, a ~ 0, and .. T(f) (t) = f[C9(t)], t E Rand fELl (lR).

.

A proof of this result can be found in [Ka, pp. 217-219]. Some further discussion of such isomorphism problems is available in [Ru l , pp. 77-96]. 5.4. Characterization of Singular Elements in Self-adjoint Semisimple Commutative Banach Algebras. In our earlier discussion of topological zero divisors we proved (Proposition 1.6.2) that, if x is a topological zero divisor in a commutative Banach algebra A with identity, then x is singular. For the special algebra C(X), X being a compact Hausdorff topological space, we sa\.; in Theorem 1.6.1 that f E C(X) is a topological zero divisor if and only if f is singular. For an arbitrary commutative Banach algebra A with identity it need not be the case that a singular element is a topological zero divisor, but if A is self-adjoint and semisimple, then it is true. We shall prove this in the theorem of this sectioR. Before we can do this, however, we need some preliminary results about self-adjoint semisimple commutative Banach algebras. Suppose A is such an algebra. Given x € A, we know that there .. .. exists some yEA such that y = x. Since A is semisimple, this element y is clearly unique. ~hese observations lead to the following definition: Definition 5.4.1. Let A be a self-adjoint semisimple commutative Banach algebra. If x ~ A, then we denote by x* the uni... que element of A such that x* = x.

.

5. Semisimp1e Commutative Banach Algebras

140

We shall need the next proposition. Proposition 5.4.1. Let A be a self-adjoint semisimple commutative Banach algebra. Then (i) x,y (A

(x + y)* and

a €

= x*

+

y*, (xy)*

= x*y*,

and

(ax)* = ax*,

~.

(ii) The mapping * : A - A, defined by * : x - x*, x E A, is an antilinear topological isomorphism of A onto itself. Proof. Part (i) and the fact that linear isomorphism are easily verified.

* is a surjective antiWe shall prove explicitly

only that * is a topological isomorphism. To this end we introduce a new norm in A by defining IIxlll = IIx*lI, x (A. It is a routine exercise to check that 11'11 1 is indeed a norm. _For example,

•

if IIxlll = 0, then IIx*1I = 0, and so x* = O. Hence x = 0, which • implies that X = 0, and so x = 0, as A is semisimple. Moreover, 11'11 1 is a complete norm on A. Indeed, suppose {xnJ is a Cauchy sequence in (A, U·I;l)' Then evidently {x*) is a Cauchy sequence in (A,II'II), and so there n

exists some yEA such that x = y*, we see that limllxn - xIII n

lim IIx* - yll n

n

= limllx*n

= O.

But then, if

x"l1

n

= limllx*n

- yll

n

=

Thus

(A,II'II I )

o.

is a Banach space.

Consequently we see that (A, Ii '11 1) and (A,II'II) are both commutative Banach algebras and (A, Ii 'Ii) is semisimple, whence we conclude, by Corollary 5.3.1, that 11'11 1 and 11·11 are equivalent norms. Thus, in particular, there exist Kl > 0 and K2 > 0 such that K2lixli < IIx* II ~ Klllxll, x E A, and so * is a topological isomorphism. c

5.4. Characterization of Singular Elements

141

One further observation is in order before we can state and prove the indicated theorem. If A is a commutative Banach algebra, then x ~ A is a topological zero divisor if and only if there {y} C A such that inf lIy II > 0 and n n n The proof of this assertion is left to the reader.

exists a sequence lim lixy II = O. n n

Theorem 5.4.1. Let A be a self-adjoint semisimple commutative Banach algebra with identity e. If x E A, then the following are equivalent: (i)

x

is a topological zero divisor.

(ii)

x

is singular.

Proof. The implication from part (i) to part (ii) is contained in Proposition 1.6.2. Conversely, suppose x E A is singular. A

Then _0 ( o(x) = R(x),

by Theorem 3.4.1.

Set

y

= xx*.

Clearly

1 = ~i = '~l2,

and so 0 C R(y) and R(y) C [O,~). Thus we see that -lIn ~ oCy), n = 1,2,3, ... , so that y + (l/n)e is regular, n = 1,2,3,.... Since lim liy + (l/n)e - yli = 0, we conclude that n y (bdy(A- 1 ), and so, by Corollary 1.6.1, y is a topological zero divisor. infn Ilyn II > 0

Hence there exists some sequence and

1imn lIyyn II

= 1imn Iixx*yn II = O.

(y) n

C

A such that

If inf IIx*y II > 0, then clearly x is a topological zero n n divisor. On the other hand, suppose inf lix*y II = O. Then there n

n

exists a subsequence (y ) of (y) such that limkllx*y II = O. nk n nk However, in view of Proposition 5.4.1, there exist Kl > 0 and K2 > 0 such that K2lizll ~ IIz*1I < KlllzlI, z E A. see that lim.!Ixy* II = O. Furthermore, k. nk

Thus, once again, we see that

x

Consequently, we

is a topological zero divisor.

U

CHAPTER 6 ANALYTIC FUNCTIONS AND BANACH ALGEBRAS 6.0. Introduction. Let A be a commutative Banach algebra with identity and suppose p is a polynomial. Then it is obvious that p(x) makes sense for any x E A and belongs to A. Moreover, the Polynomial Spectral Mapping Theorem (Theorem 2.2.1) entails • • that p(x) (T) = p[X(T)], T E ~(A). Our main concern in this chapter will be to show that such a phenomenon is valid for a much larger class of functions than polynomials. Thus, we shall see that, if x E A and f is analytic in some open set containing a(x), • • then there exists some y = f(x) E A such that f(x) (T) = f[x(T)], T e 6(A). The discussion and proof of this result will take up Section 6.1. The topics in the succeeding sections are essentially applications of the results of Section 6.1. Besides a generalization of the Polynomial Spectral Mapping Theorem, we shall establish a sufficient condition for a commutative Banach algebra with compact maximal ideal space to have an identity; prove a generalization of Wiener's Theorem (Theorem 4.6.2); show that, if f is a nonconstant entire function and x E A is such that f(x) = 0, then there exists a nonconstant polynomial p such that p(x) = 0; and give a description of the connected component of the identity in the set of regular elements in A. 6.1. Analytic Functions of Banach Algebra Elements. Let A be a commutative Banach algebra with identity, let x·e A, and suppose f is a complex-valued function defined and analytic on some open set 0 ~ a(x). Our main goal in this section will be to

142

6.1. Analytic Functions

143

show how to define f(x) as an element of A such that A A f(x) (T) = f[x(T)], T E 6(A). We shall accomplish this through a series of lemmas culminating in the desired result. Afterward we shall briefly discuss, without proof, some generalizations of the development. To begin we first need a number of elementary definitions.

a

~

Definition 6.1.1. A £omplex-valued function t ~ b, is said to be a smooth arc if (i) (ii)

dC/dt exists and is continuous on d~/dt

and dll/dt

Crt) =

~(t) + i~(t),

[a,b].

have no common zero on

[a,b].

Furthermore, C is said to be a regular curve if there exists a = to < tl < ... < t n _l < tn = b such that C restricted to [tk,t k+ l ] is a smooth arc, k = O,I,2, ••. ,n - 1. In the definition of a smooth arc we mean, of course, that the appropriate one-sided derivatives exist at t = a and t = b. Definition 6.1.2. Let C{t), a < t ~ b, be a regular curve. Then C is said to be simple if there exist no points a ~ tl < t z < b such that ,(tl) = '(t 2), and , is said to be a simple closed regular curve if C(a) = C(b) and there exist no points a < tl < t2 < b such that ,(tl) = C(t 2). The disjoint union of a finite number of simple closed regular curves is said to be a regular contour. Thus a simple regular curve , does not cross itself, and a simple closed regular curve meets itself only at its end points. Now suppose A is a commutative Banach algebra with identity and let x (A. Then o(x) is a nonempty compact subset of C, which may, however, consist of several disjoint pieces and whose complement need not be connected. Nevertheless, if 0 C C is an open set such that 0 ~ a(x), then it is not difficult to verify

144

6. Analytic Functions and Banach Algebras

that there exists a regular contour y

that satisfies the following

properties, where n(Y,a) denotes the winding number of the regular contour y with respect to the complex number a [A, pp. 114-117]: (1)

yeO - o(x).

If a E o(x), then n(Y,a) > 0; inside the regular contour y. (2)

(3)

If

a E o(x),

then

n(Y,a)

= 1;

that is,

o(x)

lies

that is, each point

a E o(X) lies inside precisely one of the simple closed regular curves that make up y. (4)

a

E~ -

If a ~ 0, then n(y,a) = 0; that is, no point 0 lies inside of the regular contour y.

Definition 6.1.3.

Let

A be a commutative Banach algebra

with identity, x E A, and suppose 0 ~ o(x) is an open set. If Y is a regular contour that satisfies properties (1) through (4) , then Y is said to be a spectral contour for

a(x)

lying in

O.

application of the Cauchy Integral Formula [A, p. 119] immediately establishes the first lemma. An

Lemma 6. 1. 1.

Let

A be a commutative Banach algebra with

identity and let

x C A.

o~ a(x)

is any spectral contour for

and

y

If f

is analytic in some open set a(x)

lying in

0,

then f(a) -- _1_ '" , f(,) d'" 2ni 'y - a ~

(a E a(x)).

We wish Rext to extend this result so that we can replace a in the above equation by x. To do this we clearly need to define the integral

where e denotes the identity of A,

and show that it belongs to

A.

6.1.Analytic Functions

145

With this in mind suppose A is a commutative Banach algebra with identity e, x E A, O~ a(x) is open, and f is analytic on 0, and let y be a spectral contour for a(x) lying in O. To make the argument a bit simpler we shall assume for the moment that is a simple closed regular curve then C ~ a(x), and so (Ce - x)-l

y

mapping sion in

C = C(t), a
-1

C - (Ce - x) is continuous from y to A, since inverA is a continuous operation (Corollary 1.4.3). Since y

is compact, we then deduce that the mapping is uniformly continuous.

C - (Ce - x)

-1

, C E y,

Let a = to < tl < ... < t n _1 < tn = b be any partition of [a,b] and set 'k = 'Ct k), k = O,1,2, ... ,n. Then considering the finite sums

we deduce, by essentially the same argument as for scalar functions [Go, pp. 52-54], that

exists in A. The details are left to the reader. obtained is denoted by f(x)

1 = 2ni

Jy

The limit so

f(C)(Ce - x) -1 dC.

Note that the argument works equally well for a spectral contour y on repeating the preceding development for each of the finitely many simple closed regular curves making up y. These observations prove the first portion of the next lemma. Lemma 6.1.2. Let A be a commutative Banach algebra with identity e and let x (A. If f is analytic in some open set o ~ a(x) and y is a spectral contour for a(x) lying in 0, then

6. Analytic Functions and Banach Algebras

146

1 Jy f(C)(Ce - x) -1 dC f(x) = 2ni

exists and is an element of A. T E 6(A) .

A

Moreover,

f(x) (T)

A

= f[x(T)],

Proof. We need only prove the final assertion of the lemma. Let T E 6(A). Then we see that, for C ~ a(x) , 1

-1

= T[(Ce

- x)(Ce - x)

= T(Ce

- x)T[CCe - x)

= [C

]

-1

]

- T(X)]T[(Ce - xl-I],

whence we deduce that T[CCe - x) -1 ] for any of the approximating sums to

- T(X)] -1 . f(x), say

= [C

Consequently,

1 n-I -1 (2ni) E f(Ck)CCke - x) (C k + I - Ck ), k=O

we have 1 n-l -1 T[G2ni ) E f(C k) (Cke - x) (C k+ 1 - Ck)] k=O

=

1 n-l -1 (2ni) 1: f(C k) [C k - T(X)] eC k+ I - Ck), k=O whence we conclude from the definition of f(x) and the continuity of T that A

f(x) (T)

= T[f(x)] = 21. n1

Sy

f(C)[C - T(X)]-1 dC

= f[TeX)] A

= f[X(T)], •

A

since X(T) E R(x)

= a(x),

by Theorem 3.4.1.

o

6.1. Analytic Functions

147

Our definition of f(x) seems to suffer from one disadvantage: the value f(x) appears to depend on the choice of the spectral contour y. That this disadvantage is illusory follows from the next lemma. Lemma 6.1.3. Let A be a commutative Banach algebra with identity e and let x € A. If f is analytic on some open set o~ a(x) and YI and Y2 are spectral contours for a(x) lying in 0, then 1 JY f(C)(Ce - x) -1 dC 2ni I

1 = 2ni

r

f(C)(Ce ~ x) -1 dC·

Jy 2

Proof. Suppose x* is any continuous linear functional on A. Then, as seen previously (for example, in the proofs of Theorems 1.5.1 and 2.1.2), we see that the mapping C - x*[(Ce - x)-l] is analytic on C - a(x). Thus it follows at once from the same sort of limiting argument as in Lemma 6.1.2 and the classical Cauchy Integral Theorem [A, p. 145] that 1 JY f(C)(,e - x) -1 dC] = 2ni 1 JY f(C)x*[(Ce - x) -1 ] dC x*[2ni I l

= 2~i

Iy

f(CJx*[(Ce - xl-I] dC 2

1 &y r = x*[2ni

f(C)(Ce - x) -1 dC]. 2

A consequence of the Hahn-Banach Theorem [L, p. 90] then implies that

Thus we see that the definition of f(x) which spectral contour is utilized. Our next lemma describes what choices of f.

f(x)

does not depend on

is for some particular

6. Analytic Functions and Banach Algebras

148

Lemma 6.1.4. Let A be a commutative Banach algebra with identity e and let x C A. Then (i)

If

f(C) = 1, ,

(.~,

then

f(x) = e.

Cii)

If

fCC) = C, C €~,

then

f(x)

= x.

(iii) If f(C) = E:=OanC n , , (~, is an entire function, then n f(x) = Ln=Oanx, where the series in A converges absolutely in norm. CD

Proof. Clearly we need only prove part (iii), and we may take o = Ii.: => o(x). In view of Theorem 3.4.1 we see that, if r > !lxII, then the circle y = (C I Ici = r) is a spectral contour for o(x). Moreover, if C (V, then, by Corollary 1.4.3 and the comments following it or by direct computation, we see that CD

= ,-lee _ X)-l = ,-1 1:: (x)n C n=O ,

(Ce - x)-l

and that the convergence is absolute and uniform in norm on V. Hence we deduce that 1 f(x) = 2ni

.

.I y

f(C)(Ce - x)

[~J fCC) dC]x n n=O 2n1 V Cn+ l

=

f(n) (0) I n= O n.

1:

1:

CD

=

dC

CD

=

n

CD

\

-1

X

n

a x , n=O n 1:

by an application of the Cauchy Integral Formula [A, p. 120].

c

It is apparent from the definition of f(x) that the mapping from f - f(x) is linear. However, more can be said, as seen in the next lemma.

6.1. Analytic Functions

149

Lenuna 6. 1. 5 . Let A be a conunutative Banach algebra with identity e and let x E A. If f and g are analytic on an open set O:Ja(x), then (i) (ii) (iii)

(f

+

= f(x)

g) (x)

= af(x),

(af) (x)

+

a

g (x). E; \l;.

(fg) (x) = f (x) g (x) .

Proof. We need only prove part (iii). Let VI and Y2 be distinct spectral contours for a(x) lying in 0 such that YI lies inside V2 , that is, such that n(v 2 ,C) > 0, , E VI' We note that, if z (Y 2 and C E Yl , then (Ce - x)-l(ze - x)-l(z - C)

= (Ce

- x)-l(ze - x)-l[(ze - x) - (Ce - x)]

= (Ce - x)

-1

- (ze -x)

-1

.

Using this observation we see that I. f(x)g(x) = [-2

J'

TTl

= (2;i)2

SY

I

1

x) -1 dz]

SV l JV2 f(C)g(z)CCe - x)-lCze - x)-1 dz dC l

- 2ni

&

1

= (2ni)2 1 J'Y JY2 1 -- 2TTi

rY2 g(z) (ze -

1 f(C) (Ce - x) -1 dC] [2n'

YI

f

-1

f(C) ( l[CCe - xl - (ze - x) g z z - ,

-1

] dz dC

1 J~ (C)(Ce - x) -1[ 2ni Y2 zg(z) _ C d z ] dC

JY

2 g(z)(ze - x)

-1

J

1 f(C) [2ni Yl z _ CdC] dz.

Now g is analytic on and inside Y2 , so for each by the Cauchy Integral Formula [A, p. 119], gCC)

C E Y1 we have,

=~ J g(z) dz 2TTI Y2 z - C

as Y l lies inside Y2' On the other hand, f(C)/(z - ') is analytic on and inside VI as a function of C for each z E Y2' whence

6. Analytic Functions and Banach Algebras

150

we conclude, from the Cauchy Integral Theorem [A, p. 145], that for each

z E Y2

1 j. f(C) dC = 0 2ni Y1 z - C •

Consequently we have f(x)g(x)

I = 2ni

SY

f(C)g(C)(Ce - x) -1 dC = (fg)(x),

1

and the lemma is proved.

0

The reader should note the crucial, if tacit, role plated by Lemma 6.1.3 in the proof of the preceding result. We can summarize the preceding development in Theorem 6.1.1. The proof of part (iii) is left to the reader. Definition 6.1.4. Let A be a commutative Banach algebra with identity and let x € A. The collection of all complex-valued functions f that are defined and analytic on some open set o ~ a(x) will be denoted by A(x). The open set 0 in the preceding definition may, of course, depend on f. Evidently A(x) is an algebra under pointwise operations of addition, scalar multiplication, and multiplication. Theorem 6.1.1. Let A be a commutative Banach algebra with identity and let x E A. Then the mapping f - f(x), f E A(x), is an algebra homomorphism from A(x) into A such that (i)

If

f E A(x) ,

then

•

•

f(x) (T) = f[X(T)], T E ~(A).

(ii) If f(C) = ~=oanCn, C E~, is an entire function, then f(x) = ~=oanxn, and the convergence is absolute in norm. (iii)

Suppose

(f) c A(x) n

and

f E A(x)

are analytic on an open

set 0 ~ a(x). If the sequence (f) converges uniformly to n each compact subset of 0, then lim IIf (x) - f(x)1I = o. n

n

f

on

6.2. Consequences of the Preceding Section

151

Portions of this theorem can be extended to the case where a(x) is replaced by the joint spectrum O(x l .x 2••••• xn) of finitely many elements of A. We shall not give a proof of this theorem since it involves the techniques of the theory of functions of several complex variables, and a complete discussion would necessitate a rather lengthy digression. We shall, however, state the indicated result. Proofs are available. for example, in [HOI' pp. 101-114; Hr, pp. 68-70; Ri, pp. 156-162; S. pp. 60-72]. Theorem 6.1.2 (~ilov-Arens-Calderon). Let A be a commutative Banach algebra with identity and let x l ,x 2 ' .•.• xn be in A. If f is a function of n complex variables that is defined and analytic on some open set 0 ~ a(x l 'x 2 ' ..•• x). then there exists some • •• n. yEA such that yeT) = f[x 1 (T),X 2 (T) •••.• Xn (T)], T E A(A). Theorem 6.1.1 also gives a partial answer to a more general question. If A is a commutative Banach algebra and f is defined • on some open set 0 c~, then we say that f operates in A if • • • fox E A whenever x E A and R(x) c O. Theorem 6.1.1 then asserts • that functions that are analytic on open sets in ~ operate in A for any commutative Banach algebra A with identity. On the other hand, if A = C(X), X being a compact Hausdorff topological space, then it is easily verified that any continuous function defined on • an open subset of ~ operates in A. The question of determining • for a particular algebra in A and we shall not pursue it here. to [Ka, pp. 235-249; Ri. p. 167;

precisely which functions operate A is in general a difficult one, The interested reader is referred Ru l , pp. 137-155].

6.2. ~ Consequences of the Preceding Section. In this section we wish to establish a number of results that are simple consequences of the development in the preceding section. In particular we shall obtain generalizations of the Polynomial Spectral Mapping Theorem (Theorem 2.2.1) and Wiener's Theorem (Theorem 4.6.2).

152

6. Analytic Functions and Banach Algebras

Theorem 6.2.1 (Spectral Mapping Theorem). Let A be a commutative Banach algebra, let x E A, and suppose f is a complexvalued function that is defined and analytic on some open set O~ a(x). Then (i)

If

A has an identity, then

f[a(x)]

= a(f(x».

(ii) If A is without identity and f(O) = 0, then there exists some f(x) E A such that f[a(x)] = a(f(x». Proof. Part (i) follows immediately from Theorem 6.l.l(i) on • and a(f(x» = R[f(x)] • sin~e A has recalling that a(x) = R(x) an identity. To establish part (ii) we need first to prove the existence of a suitable element f(x) E A. The development of the preceding section is not directly applicable, as A is without identity. However, by Theorem 2.1.1, we know that a(x) = aA(x) = aA[e] (x), so we can apply Theorem 6.1.1 to the algebra A[e] to deduce the existence • of an f(x) E A[e] such that f(x) • (T) = f[X(T)], T E ~(A[e]). However, if TCD E ~(A[e]) denotes,as usual, the unique complex homomorphism on A[e] that vanishes identically on A, we see that • ) ] = f(O) = 0 by the assumptions on f. Hence f(x) • (TCD) = f[X(T CD f(x) E A.

• The definition of f(x) also reveals that f(x) • (T) = f[x(T)], T E ~(A), which, combined with the facts that f(O) = 0, • • a(x) = R(x) U {oj, and a(f(x» = R[f(x) ] U {oj, implies that f[a(x)] = a(f(x». o A similar result is the following corollary: Corollary 6.2.1. Let suppose ~(A) is compact. and f is a complex-valued • on some open set 0 ~ R(x), •yeT) = f[X(T)], • T E A(A).

A be a commutative Banach algebra and If x E A is such that •X(T) ~ 0, T E ~(A), function that is defined and analytic then there exists some yEA such that

6.2. Consequences of the Preceding Section

153

Proof. If A has an identity, then ~(A) is automatically compact, and the corollary follows at once from either Theorem 6.l.l(i) ,. or 6.2.1. So assume that A is without identity. Since x is ,. continuous, we see that R(x) is a compact subset of ~ that does not contain zero. Thus there evidently exist open sets 01 and 02 in q; such that

,. (a)

R(x)

C

(b)

0 €

02.

(c)

01 n 02

01

C

0.

= ~.

Having chosen such sets, define g(C)

= f(C)

g(C) = 0 for

for

C ~ 01'

C E 02.

Clearly g is analytic on the open set

01 U 02

~

a(x),

and

g(O) = O.

Consequently, by Theorem 6.2.l(ii), there exists some y = g(x) ,. ,. in A such that yeT) = g[X(T)), T E ~(A), from which it follows ,.,. ,. at once that yeT) = f[X(T)), T E ~(A), as R(x) cOl· 0 As we know from Theorem 3.2.2, if A is a commutative Banach algebra with identity, then ~(A) is compact. It is natural to ask ~hether or not the converse assertion is valid. Some support for such a conjecture is given by the next result. CorGllary 6.2.2. Let A be a semisimple commutative Banach algebra. If ~(A) is compact and there exists some x E A such ,. that X(T) ~ 0, T E A(A), then A has an identity.

,.

Proof. Since R(x) is a compact set in ~ that does not con" such that 0 ~ o. tain zero, there exists an open set ~ R(x) Then f(,) = I'" , E 0, is analytic on 0, and so, by Corollary

°

154

6. Analytic Functions and Banach Algebras

• • • 6.2.1, there exists some yEA such that yeT) = f[X(T)] = l/X{T). T ~ A(A). It is then immediate from the semisimplicity of A that xy is an identity for A. o

Actually. one can prove that, if A is a semisimple commutative Banach algebra and A(A) is compact, then A has an identity. The proof. however, requires the use of the theory of functions of several complex variables and will not be discussed here. The interested reader is referred to [Ri, pp. 167-169; 5, pp. 75-77]. An appropriate analog of the §ilov-Arens-Calderon Theorem (Theorem 6.1.2) for algebras without identity and the §ilov Idempotent Theorem [Ri, p. 168; 5, p. 73] are the main tools needed to establish the indicated result. The semisimplicity of A is necessary for the validity of the result. Another easy application of Theorem 6.1.1 yields the following generalization of Wiener's Theorem (Theorem 4.6.2): "t

Theorem 6.2.2 (Wiener-Levy Theorem). Let f = (e l I -n < t < n). If h E AC(f) and f is a complex-valued function defined and analytic on some open set 0 ~ R(h), then there exists some g € AC(f) such that g = f 0 h. Proof. The result follows immediately from Theorem 6.I.l(i) on recalling that the Gel'fand transformation on AC(f) is the identity mapping of AC(f) into C{f) (Theorem 4.6.1). Wiener's Theorem is, of course, the special case of this result it where h(e ) ~ 0, -n ~ t < n. and f(C) = lIe. , E ~. 6.3. Zeros of Entire Functions. As another application of the results of Section 6.1 we shall show that, if x is an element of a commutative Banach algebra A with identity and f is a nonconstant entire function such that f{x) = 0, then there exist some nonnegative integer nand bk €~, k = 0,l,2, ... ,n such that

6.4. Connected Component of the Identity tk=Obkxk = 0,

155

where not all of the bk are zero.

Theorem 6.3.1. Let A be a commutative Banach algebra with identity and let x (A. If there exists a nonconstant entire function f such that f(x) = 0, then there exists a nonconstant polynomial p such that p(x) = o. Proof. From Theorem 6.1.1 we know that f(x) E A for any • • entire function f and that f(x) (T) = f[x(T)], T E ~(A). Hence, • since f(x) = 0 by assumption, we see that f[x{T)] = 0, T E A(A); that is, f vanishes identically on a(x). Since a(x) is compact and the zeros of a nonconstant entire function are isolated [A, p. 127], we conclude that a(x) is finite, say

Let mk denote the multiplicity of the zero of f k = 1,2, .•• ,r, and define

at

~,

r

p(C) =

n

(C - a )mk. k=l k

Then, as is easily verified, the function g defined by gCC) = fCC)/pCC), C (C, is an entire function that never vanishes • on o(x); that is, g[X(T)] ~ 0, T ~ 6(A). However, by Theorem 6.1.1 and Corollary 3.4.2(ii), we see that g(x) (A is regular and f{x) = p{x)g{x) = O. Therefore p(x) = 0. 0 6.4. The Connected Component of the Identity in A-I. If A is a commutative Banach algebra with identity e, then we have seen previously that A-1 , the set of regular elements in A, is an open subgroup of A. We now wish to show, at least in the case that A is semisimple, that the connected component of e in A-I is precisely the set of those elements of A of the form ~=oxn/n!, x E A. We recall that the connected component of point in a topological space is the largest connected set that contains the given point.

6. Analytic Functions and Banach Algebras

156

In what follows, exp will denote the entire function m n exp(C) = ~n=O' In!, C (Cj that is, exp is the complex exponential function. We need two lemmas before the indicated theorem. The proof of the first lemma is left to the reader. Lemma 6.4.1. Let A be a semisimple commutative Banach algebra with identity. If x,y € A, then (i) (1• 1· )

(iii)

exp (x) € A. w» n l n .. I exp ( x) -- Ln=OX

exp(x

+

y) = exp(x)exp(y).

Lemma 6.4.2. Let A be a semisimple commutative Banach algebra with identity e. If x E A and lie - xII < 1, then there exists some y (A such that exp(y) = x. Proof. Since lie - xII < 1, we see from Theorem 1.4.1 that • is a compact x is regular, whence, by Corollary 3.4.2{ii), R(x) set that does not contain zero. Let 0 be an open subset of C • such that 0 ~ R(x) and 0 ~ O. Then fCC) = 10gC, C € 0, is analytic on 0, and so, by Theorem 6.1.1, y = f(x) E A and yeT) = log X(T), T (6(A). But exp{y) E A and

..

.

• exp{y) • (T) = exp[Y(T)] = exp[log • X(T)] • = X(T) from which we conclude that exp(y)

= x,

as

(T

E 6(A»,

A is semisimple.

o

We can now state and prove the desired theorem. Theorem 6.4.1. Let A be a semisimple commutative Banach algebra with identity e and let exp(A) = (exp(x) I x € Al. Then exp(A) is the connected component of e in A-1 . Proof. From Lemma 6.4.1 we see that exp(x)exp(-x) = exp{O) = e, -1 x ~~ A, and so e ~~ exp(A) cA. Moreover, if x E A, t h en"1t lS easily verified that wet) = exp{tx), 0 < t < 1, defines a continuous

6.4. Connected Component of the Identity

157

curve in exp(A) from e to exp(x), from which it follows at once that exp(A) is connected. Thus exp(A) is contained in the connected component of e in A-1 . To show that exp(A) is this connected component it suffices to prove that exp(A) is both open and closed in A-I.

IIY -

To this end suppose y = exp(x) zll < lIlly-III. Then lie - y-l zl1

and let

= lIy-l y

z (A be such that

_ y-l zl1

~ lIy- 111l1y -

zll

< 1,

and so, by Lemma 6.4.2, there exists some w E A such that exp(w) = y-I z. Consequently

z whence

z

~

= y[exp(w)] = exp(x)exp(w) = exp(x

exp(A),

and

exp(A)

+

w),

is open.

On the other hand, suppose y E A-I and y E cl[exp(A)]. Then there exists some z = exp(x) such that lIy - zll < lIlly-III. Arguing as before, we conclude that y-I Z (exp(A), and so y-l E exp(A). Thus y E exp(A) and exp(A) is closed. Therefore exp(A)

is the connected component of

e

in A-1 0

Actually the theorem and the lemmas are valid in any commutative Banach algeb~a with identity. We have proved the results only for semisimple algebras because the arguments in Lemmas 6.4.1(iii) and 6.4.2 are less technical in this case. The lemmas can be established in the general case via a power series argument. See, for example, [8, pp. 49 and 50]. We shall not pursue the discussion of A-I and exp(A) any further, but content ourselves with mentioning one of the most

158

6. Analytic Functions and Banach Algebras

important theorems on the subject. a subgroup of A-I.

Note that

exp(A)

is evidently

Theorem 6.4.2 (Arens-Royden). Let A be a commutative Banach algebra with identity. Then the quotient group A-1/exp(A) is isomorphic to HI[a(A),Z], the first ~ech cohomology group of a(A) with integer coefficients. Using this result one can prove, for instance, that A-1 is either connected or has an infinite number of connected components. Discussions of the Arens-Royden Theorem and some of its consequences can be found in [Ga, pp. 88-91; 5, pp. 98-104; Wm 2 , pp. 88-96]. More generally, other treatments of the role of analytic functions in the study of Banach algebras are available in [B, Ga, Ho l , Hr, Lb, Ri, 5, wm l , Wm 2].

CHAPTER 7 REGULAR COMMUTATIVE BANACH ALGEBRAS 7.0. Introduction. If X is a compact Hausdorff topological space, then it is well known that X is a completely regular topological space and a normal tOP91ogicai space. The first assertion is equivalent to saying that for each closed set E C X and each point t E X, t t E, there exists some f E C(X) such that < f(s) < 1, s ( X, fet) = 1, and fes) = 0, sEE. The second assertion, combined with Urysohn's Lemma, reveals that, if El C X and E2 C X are disjoint closed sets, then there exists some f E C(X) such that 0 ~ f(s) ~ 1, s ( X, f(s) = 1, s (E l , and f(s) = 0, s E E2 . Recalling that the Gel'fand representation of • = C(X), we see the commutative Banach algebra A = C(X) is just A • that these observations say precisely that A has the indicated properties. In particular, if A = C(X), X being a compact Haus• separates disjoint closed subsets dorff topological space, then A of A(A). Our goal in this chapter is to determine sufficient conditions on an arbitrary commutative Banach algebra A such that the • possess separation properties analogous to those elements of A described here.

°

In order to accomplish this we shall introduce a new topology on ~(A), the hull-kernel topology, and show that, when this topology coincides with the Gel'fand topology on A(A) , then, whenever E C A(A) is closed and T E A(A), T ~ E, there exists some x E A • • such that X(T) = 1 and xew) = 0, wEE. Furthermore, in this case, we shall also see that, if K C A(A) is compact, E C A(A) is closed, and

• X(T) = 1,

T E K,

KnE and

=~,

•X(T)

then there exists some

= 0,

T

~

159

E.

x

E A such that

7. Regular Commutative Banach Algebras

160

A commutative Banach algebra A that satisfies either of the foregoing separation properties will be termed regular or normal, respectively.

The hull-kernel topology and regular commutative

Banach algebras play a central role in the study of the ideal structure of commutative Banach algebras, as we will see in the next chapter. 7.1. The Hull-Kernel Topology and Regular Commutative Banach Algebras. The main concerns of this section will be to define and investigate the elementary properties of the hull-kernel topology on A(A) and to determine when this topology coincides with the Gel'fand topology. We begin with a number of preliminary definitions.

EC

Definition 7.1.1. Let A be a commutative Banach algebra. If A(A) , then the kernel of E, denoted by keEl, is defined as k(l:)

= n

M

=

MEE if E ~~, while k(~) = A. If I C A is an ideal, then the hull of I, denoted by h(I), is defined as h(l)

= [M I M £ A(A), = (T I

For each x E A, as

M~ IJ

T E A(A), T-I(O) ~ IJ.

the

~

set of x,

denoted by Z(x),

Z(x)

= {T

T E. A(A), X(T)

= {M I

ME A(A),

..

x

whereas if I C A is an ideal, then the by Z(I), is defined as Z(I) =

fl Z(x) = {T

= 0)

EM), ~

set of

..

I,

T (A(A), X(T) = 0, x ( I)

xEI

= {M

is defined

M E a(A), I eM).

denoted

7.1. Hull-Kernel Topology

161

It is evident from the definitions and the continuity of x• that Z(x) and Z(I) are closed subsets of ~(A), and Z(I) = h(I). Moreover, we have the following proposition: Proposition 7.1.1. (i)

If I

(ii)

heAl

(iv)

If E C

keEl

= (x I

Let A be a commutative Banach algebra.

A is an ideal, then h(l)

C

=~

and h«(O)

~(A),

= h[cl{I)].

= ~(A).

then

..

= {x I

x € A, X(T) = 0, TEE) then keEl

x € A, Z(x) ~

(v)

If E C

~(A),

(vi)

k[~(A)]

= Rad(A).

(vii)

If I

C

A is an ideal, then cl(I)

(viii) h = hkh.

If I

C

A is an ideal, then h{I) = h(k[h(I)]);

is a closed ideal in A.

C

k[h{I»).

(ix)

If E C

~(A),

then keEl = k(h[k(E)]);

(x)

If E C

~(A),

then h[k(E)]

(xi) If El C ~(A), E2 and h[k(E l )] C h[k(E 2)]. (xii)

If El

C ~(A)

C ~(A),

and E2 c

El.

~

k

= khk.

E.

and El

~(A),

that is,

that is,

C

E2,

then k(E l )::> k (E 2)

then h[k(E l ) n k(E 2)]

=

h [k (E l U E2 )].

Proof. All of the proofs are rather elementary. only one proof and leave the rest to the reader. Suppose h (k [h (I) ] ) •

We shall give

I C ~(A) is an ideal. We shall show that h(l) = If M E h(k[h(I»)), then M~ k[h(I)] ~ I by part (vii),

7. Regular Commutative Banach Algebras

162

whence M E h(I). Conversely, suppose T ~ h(k[h(I»)) = Z(k[h(I»)). " Then there exists some x E k[h(I)] such that X(T) ~ 0, while A from part (iv) we see that x(w) = 0» W E hel). Hence T ~ h(I)" and so h(l) = h(k[h(I)]), which proves part (viii).

o

As the reader should guess from the preceding proof, in discussing hulls and kernels it is often of considerable advantage to keep in mind the descriptions of these objects both in terms of maximal regular ideals and in terms of complex homomorphisms. The comment preceding Proposition 7.1.1 shows that h(l) is a closed subset of 6(A) for each ideal leA. The idea behind the hull-kernel topology is to use such closed sets, that is, hulls of ideals, as the closed sets in a topology. With this in mind we make the following definition: Definition 7.1.2. Let A be a commutative Banach algebra. If E c 6(A) , then the hull-kernel closure of E, denoted by E, is defined to be E = h[k(E)]. The reader should recall that the closure of E in the Gel'fand topology is denoted, as usual, by cl(E). In order to show that the hull-kernel closures of sets in 6(A) are actually the closed sets for some topology on 6(A) we must show that the correspondence E - E, E C 6(A) , is a closure operation [DS l , pp. 10 and 11]; that is, we must show that the operation of forming the hull-kernel closure satisfies the following conditions:

= 6(A).

(a)

6(A)

(b)

E

(c)

E = E.

Cd)

(E l U E2 )-

C

E.

= El

U E2 •

7.1. Hull-Kernel Topology

163

Evidently conditions (a) and (b) hold by Proposition 7.1.I(vi) and (x), respectively, and

E = h[k(h[k(E)])] = h[k(E)] = E, by Proposition 7.1.1(ix). So suppose EI we see that

C

A(A)

and

E2

C

A(A).

From Proposition 7.1.I(xi)

(k = 1,2),

whence El U E2 C (E I U E2)-. Conversely, suppose M belongs to and let u be an identity modulo M. Clearly ~(A) - (E I U E2) k(E k ) ~ M, k = 1,2, and u ~ M, 9y Proposition l.l.l(i). Thus, since M is the null space of a nonzero continuous linear functional and hence is of codimension one [L, p. 68], we see that there exist v,w ~ M, x e keel)' and y e k(E 2) such that u = x + v = y + w. 2 2 However, u - xy = u - (u - v,(u - w) = (v + w)u - VW, which 2 shows that u - xy € M, as M is an ideal, while xy E keEl) nk(E 2). Now, if M E (E I U E2) = h[k(E l U E2)], then, from Proposition 7.1.1(xii) we see that M € h[k(E 1) n k(E 2)]. Thus, in this case, we conclude that xy € M, whence u2 E M, as u 2 - xy € M. However, since u is an identity modulo M, we have u2 - u € M, and so u E M, which is a contradiction. Therefore M ~ (E I U E2) and (E I U E2)- = El U E2 . Thus we see that the correspondence E - E, E C ~(A), is a closure operation and so it can be used to define a topology on

~(A).

Definition 7.1.3. Let A be a commutative Banach algebra. Then the topology on ~(A) determined by the closure operation E - E = h[k(E)], E C ~(A), is called the hull-kernel topology. The relation of the hull-kernel topology to the Gel'fand topo109)' on ~(A) is contained in the next theorem. We recall that a topology is said to be Tl if every singleton set is closed.

7. Regular Commutative Banach Algebras

164

Theorem 7.1.1. Let A be a commutative Banach algebra. Then the hull-kernel topology on A(A) is a Tl topology which is weaker than the Gel'fand topology. Moreover, if A has an identity, then A(A) in the hull-kernel topology is compact. Proof. Since h[k([M))] = (M), M (A(A), and h[k(E)], E C A(A), is always closed in the Gel'fand topology, we see at once that the hull-kernel topology is Tl and weaker than the Gel'fand topology. A bit more work is required to show that 6(A) is compact in the hull-kernel topology when A has an identity. If A has an identity e, then in order to show that deAl is compact in the hull-kernel topology it suffices to prove that, if (E) C A(A) is a family of hull-kernel closed sets with the a finite intersection property, then n E #~. Evidently, if (E) aa a is such a family of sets, then E # ~ and k(E) is a proper a a closed ideal in A. Let I denote the closed ideal in A generated by U k(E); that is, I is the norm closure of the linear subspace a a in A generated by Uak(Ea'. It is easily seen that I is a closed ideal, and, moreover, I is proper. Indeed, if I = A, then there would exist a l ,a2 , ••. ,a and xk (k(E ), k = 1,2, ... ,n, such ~

n

~

that lie -1:k=lxkl: < 1. Hence, by Theorem 1.4.1, ~=lxk is regular. Let y be the inverse of Ik=IX k . Clearly e = Ik=IXkY and xky E k(E Ok ), k = 1,2, ... ,n. Thus we see at once that the norm closed ideal J generated by U~=lk(EOk) is all of A, as e € J. We claim that ~=lh[k(Eax)] =~. If this were not so, there would exist some M E A(A) such that M~ k(E ), k = 1,2, ... ,n, and ~ hence M~ J = A. This, ho~ever, contradicts the properness of M. Thus (~=lh[k(ECZk)] = D, whence r~=IECZk = D since EG\ = EOk = h[k(E )], k = 1,2, ... ,n. The latter assertion is contrary to the ~ assumption that (E) has the finite intersection property, and so a we conclude that I is proper.

M~

Hence, by Theorem 1.1.3, there exists some M E A(A) such that I ~ k(Ea' for each a. Consequently from Proposition 7.1.1(iii)

7.1. Hull-Kernel Topology we see that M= heM) c Therefore a (A)

nah[k(Ea)]

165

=

naa E,

whence

naa E

~~.

is compact in the hull-kernel topology.

o

The next most obvious question to raise is: ~~en do the hullkernel and Gel'fand topologies on ~CA) coincide? From Theorem 7.1.1 and an argument used several times previously we see that, if A has an identity, then the topologies will coincide precisely when the hull-kernel topology is Hausdorff. This observation is actually true in general, as is shown by the next result. Theorem 7.1.2. Let A be a commutative Banach algebra. the following are equivalent:

Then

(i) The hull-kernel topology and the Gel'fand topology on a(A) coincide. (ii)

The hull-kernel topology on a(A)

is Hausdorff.

(iii) If E c a(A) is closed in the Gel'faBd topology and A T € a(A), T ~ E, then there exists some x E A such that X(T) = I A and x(w) = 0, wEE. Proof. Obviously part (i) implies part (ii), and from the remark preceding the theorem we see that part (ii) implies part (i) if A has an identity. If A is without identity, then consider the algebra A[e]. By the comment following Definition 3.2.2 and Theorem 3.2.2(ii) we have a(A[e]) = a(A) U (T), where TCD E a(A[e]) co is the unique complex homomorphism of A[e] such that Tco (x) = 0, X (A. Denoting hulls and kernels computed with respect to aCA[e]) by hand k, respectively, it is easily verified that e

e

h [k (E)]

e

e

= h(k[E n 6(A)])

U (T co)

(E

c:

a (A [ e ]) ) .

From this it is apparent that the hull-kernel topology on a(A) coincides with the relative hull-kernel topology inherited from A(A[e]), and the same is true of the Gel'fand topology by Theorem

166

7. Regular Commutative Banach Algebras

3.2.2(ii). Since the hull-kernel topology on A(A) is assumed to be Hausdorff, it follows that the hull-kernel topology on A(A[e]) is Hausdorff, whence, as before, the hull-kernel and Gel'fand topologies on A(A[e)) coincide. Hence these topologies on A(A) also coincide, and part (ii) implies part (i) in the case that A is without identity. Now suppose the hull-kernel and Gel'fand topologies on A(A) coincide and let E CA(A) be closed and T E A(A) - E. Since E = E = h[k(E)], we see that T l h[k(E)], and so there exists • some x E keEl such that X(T) ~ O. However, since x € keEl, • = 0, wEE, and part (i) implies part (iii). x(w) Conversely, suppose part (iii) holds. Thus, if E C A(A) is closed in the Gel'fand topology and T E A(A) - E, then there exists • • = 0, wEE. Hence some x E A such that X(T) = 1 and x(w) x E keEl and T ~ h[k(E)] = E, from which we deduce that, if T ~ E, then T ~ E; that is, if E is closed in the Gel'fand topology, then E is closed in the hull-kernel topology. Consequently the Gel'fand topology is weaker than the hull-kernel topology, which, combined with Theorem 7.1.1, shows that the two topologies coincide. Therefore part (iii) implies part (i), and the proof is complete. O Recalling the discussion in the introduction to this chapter • of a commutative Banach we see that the Gel'fand representation A algebra A separates points and closed sets precisely when the hull-kernel topology is Hausdorff. We wish to single out such algebras with a special name. Definition 7.1.4. Let A be a commutative Banach algebra. Then A is said to be regular if the hull-kernel topology on A(A) is Hausdorff. In view of the topological remarks in the introduction the reader may wonder why we do not call such Banach algebras completely

7.2. Some Examples

167

regular instead of regular. Actually this is done by some authors (see, for instance, [Ri, pp. 83-96]), but the term "regular" seems to be more widely used, and for this reason we prefer it. Before we show that the Gel'fand representation of a regular commutative Banach algebra also separates compact sets from closed sets, we shall consider the question of regularity for several specific algebras. 7.2. Some Examples. As mentioned in the introduction, every compact Hausdorff topological space X is normal, whence we see that the commutative Banach algebra C(X) is regular as the Gel'fand transformation on C(X) is just the identity mapping (Theorem 4.1.1 and Corollary 4.1.1). Similarly, if X ~s a locally compact Hausdorff topological space, then C (X) is a regular algebra. To see o this, in view of Theorem 7.1.2, we need only show that, if E C X = ~(Co(X)) is closed, then it is closed in the hull-kernel topology; that is, E = h[k(E]). Denoting the o~e-point compactification of X by X+, we know that C (X)[e] = C(X+), and so, by the regularity + 0 of C(X), we deduce that E

= cl(E)

=

cl (E) n X e

=

he[ke(E)] n X

=

(h[k(E)] U {Tm )) n X

= h [k (E)] ,

where cle(E) denotes the closure of E in X+. Thus Co (X) is regular. Using Theorem 7.1.2(iii), it is also easily seen that, if a,b ~~ a < b, then Cn([a,b]), n = 1,2,3, .•• , is regular. On the other hand, the commutative Banach algebra

A(D) is not regular. Indeed, consider the set E = {lIn I n = 2,3, ..• ). Evidently E C D and cl(E) = E U (oj. But recalling that the Gel'fand transformation on A(D) is again the identity mapping,

7. Regular Commutative Banach Algebras

168

we see that keEl

= (f I =

f E A(D), f(l/n)

= 0,

n

= 2,3, •.• }

(0),

since the zeros in an open set of a nonconstant analytic function are isolated [A, p. 127]. Hence E = h[k(E)] = herO)) = A(A(D)) = D ~ cl(E). Thus the Gel'fand and hull-kernel topologies do not coincide for A(D), and so A(D) is not regular. As our final example we wish to show that LI(G), G being a locally compact Abelian topological group, is regular. However, this result is not as simply proved as the previous assertions. In order to do it we shall have to appeal to another fundamental theorem of harmonic analysis, Plancherel's Theorem, whose proof will not be given until Section 10.4. Before stating this result we wish to remind the reader of some of the discussion in Section 4.7. We saw there that the maximal ideal space A(L I (G)) is identifiable with the dual group G of G, that is, with the group of continuous homomorphisms of G into r = {, I , (Q;, 1,1 = I). Moreover, with the Gel'fand topology, G is even a locally compact Abelian topological group, so, in particular, we can speak of Haar measure ~ on G. Furthermore, the Gel'fand transform, which in this instance we refer to as the Fourier

..

..

..

transform, is defined by

..fey)

= SG

f(t) (t,y) dArt)

(y

.

E G; f E L1 (G)).

.

.

Plancherel's TheoTem assures us of the existence of a dense linear subspace Vo of L2 (G) such that Vo C LI(G) n L2 (G) , Vo C L2 (G), and on which the Fourier transform is an L2 isometry. More precisely we have the following theorem:

7.2. Some Examples

169

Theorem 7.2.1 (Plancherel's Theorem). Let G be a locally compact Abelian topological group and let A be a given Haar measure on G. Then there exists a Haar measure ~ on G and a linear subspace V of L2 (G) such that 0

.

(i)

Vo

(ii)

V

0

C

LI(G)

n L2 (G).

is norm dense in

L2 (G) .

is norm dense in

• L2 (G).

,. (iii)

V

0

,.

(iv)

IIfll2 = II f 1i 2 , f E V0 .

..

..

.

(v) The mapping f - f, f E Yo' from V0 to V0 can be uniquely extended to a linear isometry of L2 (G) onto L2 (G) . The extensiop of the Fourier transformation on Vo to all of L2 (G) will be called the Plancherel transformation, and we shall ,. denote the Plancherel transform of f once again by f. Due to the way the Plancherel transform is defined, it shares a number of the formal properties enjoyed by the Fourier transform. In particular, it is not difficult to show that, if f E L2 (G), then &,. ,.,. ,. fey) = f(-y) and [(.,w)f] (y) = fey - w), y,w E G, where, of course, the identities are interpreted to hold only almost everywhere with respect to 11. The details are left to the reader. As we shall subsequently see, the proof of Plancherel's Theorem is a nontrivial matter. However, in the case of compact Abelian topological groups the result is an easy consequence of our discussion in Section 4.7 and standard results of Hilbert space theory. Indeed, if G is compact Abelian and Haar measure A on G is normalized so that A(G) = 1, then from Corollary 4.7.4 we see that ,. G is a complete orthonormal set in the Hilbert space L2 (G). By ,. Corollary 4.7.5 the dual group G is discrete, and we may assume that Haar measure ~ on G is normalized so that ~((y}) = 1, Y (G. Then, since L2 (G) C LICG), we see that the Fourier transformation is defined on all of L2 (G), and, by a standard theorem

.

..

7. Regular Commutative Banach Algebras

170

of Hilbert space theory [L, p. 405], we have IIfll2 = [ t.lf(Y)12]1/2 yEG =

[Sa

If(y)1 2 dTJ(y)]1/2

• = IIfll2 Moreover, the Riesz-Fischer Theorem [L, p. 404] shows that the mapping f - •f, f E L 2 (G), is surjective, and so Plancherel's Theorem is proved in the case of compact Abelian topological groups. Here, of course,

Vo

= L2 (G).

OUr concern now, however, is not to prove Plancherel's Theorem, but to use it. The first consequence of the theorem we wish to mention is the next corollary. Corollary 7.2.1. (Parseval's Formula). Let G be a locally compact Abelian topological group and let A be a given Haar measure • is so chosen that Plancherel's on G. If Haar measure ~ on G Theorem is valid, then

SG Proof.

f(t)g(t) dA(t) =

Sa

•f(y)g(y) • dTj(y)

Apply Plancherel's Theorem to the identity

o The next consequence we need -- one that is of considerable interest in its own right -- says that the Gel'fand representation of Ll(G) is precisely L2 (G) * L~CG); that is,. f E Ll(G) if and only if there exist g,h E L 2 (G) such that f = g * h. Note that the equality here holds pointwise since both f• and g * h belong to Co(G).

_.

..

-

Theorem 7.2.2. Let G be a locally compact Abelian topological group and let A be a given Haar measure on G. If Haar measure ~ • is so chosen that Plancherel's Theorem is valid, then on G • • • LICG) = L2 (6) * L2 (6).

7.2. Some Examples

171

Proof. If f E Ll(G), then we can write f = glh l , where gl,h l E L2 (G). For instance, define hI and gl almost everywhere as hl(t)

= If(t)1 1/ 2 ,

= e-iargf(t)lf(t)ll/2.

gl(t)

Let g and h

A

in L2 (G) denote the Plancherel transforms of gl and hI' respectively. Then, by Parseval's Formula and the observations following A Plancherel's Theorem, we have, for each w E G,

IG = IG

A

few) =

=

Ia

f(t) (t,w) 4A(t) gl{t)h1(t) (t,w) dArt) g(y)h(w - y) d~(y)

* hew).

= g

A

Conversely, suppose g,h € L2 (G) such that the Plancherel transform of is h. Evidently f = gl h1 E Ll(G), A ceding argument reveals that f = g * Therefore

LI(G)

..

A

=

and let gl,h 1 (L 2 (G) be gl is g and that of hI and a repetition of the preh.

A

L2 (G) * L2 (G).

The regularity of L1 (G) theorem.

o

will be a corollary of the next

Theorem 7.2.3. Let G be a locally compact Abelian topological group, let A be a given Haar measure on G, and suppose Haar A measure ~ on G is so chosen that Plancherel's Theorem is valid. A A If KeG is compact and U eGis measurable and such that o < ~(U) < GO, then there exists some f € L1 (G) such that (i) (iiJ (iii)

A

= 1,

fey)

Y E K.

A

fey) = 0, ylK+U A

o ~ fey)

A

~

1, Y E G.

- u.

7. Regular Commutative Banach Algebras

172

..

Proof. Clearly Xu/ij(U) and XK_U belong to L2 (G) , where, as usual, XE denotes the characteristic function of E. Hence, by Theorem 7.2.2, there exists some f (LI(G) such that

.f = [Xu/Tj(U)]

* XK- U. Thus we see that • fey)

1 = ~(U)

=

Hence, if y E K,

faxK-u(y - w)Xu(w) d~(w)

~tU) Ju xK-U(y

then y - w

~

.. I fey) = Tl{U)

..

(y ~ G).

- w) dl1(w)

K - U, w E U,

J"U dTj(w)

and so

= I,

..

..

..

whereas, if Y ~ K + U - U, then y - w ~ K - U, w E U, whence fey) = O. Finally, it is obvious that 0 ~ fey) ~ 1, y E G.

.

..

o

Corollary 7.2.2. Let G be a locally compact Abelian topological group. If KeG is compact and WeGis open and such that W:J K, then there exists some f E LlCG) such that (i) (ii) (iii)

..fey)

..fey) 0

=

..

~ f

1, Y E K•

= 0, (y)

y

~

w.

..

5 1, Y £ G.

Proof. In view of Theorem 7.2.3 and the fact that the Haar measure of a nonempty open set is positive (Proposition 4.7.3), it suffices to find an open neighborhood U of y in G such that o .. K + U - U C W. As usual, y denotes the identity of G. To do o this we obviously may assume, without loss of generality, that Yo is in K. Moreover, we claim that it suffices to show only that there exists some open neighborhood V of y in G for which o

..

.

K + V C W.

Indeed, suppose such a neighborhood V exists. Then, by the continuity of the group operations, for each y E V there exists • such that U C V and an open neighborhood Uy of y0 in G y

7.2. Some Examples

173

Y + U c V. Set U = U (VU. Clearly U is an open neighborhood Y. Y Y of y in G and U C V. Furthermore, o

U C V = U (y yE.V whence we conclude that as we wished to show.

+

U) c V+ U U y y(V y

U - U C V.

Thus

= V + U,

K + U - U C K + V C W,

Hence it remains only to prove the existence of an appropriate neighborhood V. Again from the continuity of the group operations one deduces for each y E. K the existence of an open neighborhood • such that y + V + V C W. Evidently the family Vy of y0 in G y y {y + Vy lyE K} forms an open covering of K, and so, since K is compact, there exist v1.v 2•...• yn in K such that K c k l(Y k + V ). Let V = nP IVy' Then V is an open neighbor= Yk. °K= k hood of Yo in G, and if y E K, there exists some k, 1 ~ k < n, such that Y (Y k + VYk ' Thus

ur

Y + V C Yk

+

V Yk

from which we conclude that

+

V C Yk

K+ VC

+

w.

V Yk

+

V C W, Yk

o

Corollary 7.2.3. Let G be a locally compact Abelian topological group. Then Ll(G) is a regular commutative Banach algebra.

• • - E. Then K = {w} Proof. Suppose E eGis closed and w E. G is compact, and there exists an open neighborhood W of w such that Wn E = " as E is closed. By Corollary 7.2.2 there exists some f E LI(G) such that •few) = 1 and •fey) = 0, Y E E, whence, by Theorem 7.1.2, we conclude that L1(G) is regular. 0 For particular groups G it is possible to prove the regularity of LI(G) by direct arguments, thereby avoiding the use of Plancherel's Theorem. For example, suppose G = ~ If E c~ is closed l E. then let 0 6 > 0 be such that (t - 6,t + 6) n E = ,. and t0 0 Define h to be the continuous tent function

7. Regular Commutative Banach Algebras

174

h{s)

1 = 2n

(o + s)

for

-6 < s < 0,

h{s)

1 = -2n

(o - s)

for

o<

h{s)

=0

for

s < 6,

lsi > 0,

and set feu) = e f E LlOR),

Then

it u 0

~

J-~

Hence

that

Ll (IR) 7.3.

ius

ds

f{t) = 6 ; 0 o is regular.

•

and

f{t)

= 0,

tEE,

Normal Commutative Banach Algebras.

(u

.

and direct computation reveals that

A

t E IR.

h{s)e

f{t)

E lR).

= h{t

- to)'

which shows

In this section we

shall show that the Gel'fand representation of a regular commutative Banach algebra A separates not only points and closed sets in ~(A) but also compact sets from closed sets. Before we can prove this result, however, we need to establish some facts about the Gel'fand representation theory of ideals and quotient algebras. Once again these results are of considerable interest by themselves. Theorem 7.3.1.

I

let

C

Let

A be a commutative Banach algebra and

A be a closed ideal. Then

(i)

a(l)

is homeomorphic to

a(A) - h{I).

o

(ii) If x E I and x denotes the Gel'fand transform of x o as an element of the commutative Banach algebra I, then x is the

· 0 to 6(A) - h(I); that is,· x = xI 6 {A)-h{I).

restriction of x o

o

(iii)

If.1

•

denotes the Gel'fand representation of

I,

then

I = Il~{A)_h(I). (iv)

6(A/I) If

(v)

A/I,

ex

+

then o

I)

•

is homeomorphic to o

h(I).

(x + I) denotes the Gel'fand transform of (x + I) o is the restriction of x- to hel);

= xih(I)'

x + I

in

that is,

7.3. Normal Algebras (vi) If o then (A/I) (vii)

A/I

(A/I)

•

o

175

denotes the Gel'fand representation of A/I,

= Alh(I)' is semisimple if and only if

I

= k[h(I)].

Proof. Suppose T E ~(A). Then it is evident that T restricted to I is either a complex homomorphism on I or is identically • zero. But T(X) = 0, x E I, if and only if X(T) = 0, x E I, that is, T E h(I). Thus every T E A(A) - h(l) defines a complex homomorphism on I via restriction to I. On the other hand, suppose T (A(I). Then there exists som~ y E I such that T(Y) = 1. Define w: A - C by w(x) = T(xy), x E A. Clearly w is linear, and if x,z E A, then w(xz)

= T(XZY) = T(XZY)T(Y) = T(XYZY) = T(xy)TCZY) = w(x)w(z),

whence w E ~(A). Obviously w(x) = T(X), x E I, and w ~ h(I). Moreover, it can be shown that w is unique: Suppose w' E A(A) is such that w'(x) = T(X), x E I. Then for any x E A we have w'(X)

= W'(X)T(y) = w'(x)w'(y) = w'(xy) = T(xy) = w(x),

and so w' = w. Thus we see that the restriction of T E A(A) - h(I) to I defines a bijective mapping between A(I) and A(A) - h(I). Furthermore, from the definition of this mapping it is apparent that

o

•

0

x = xIA(A) -h(I)' x E I, where x denotes the Gel'fand transform of x E I. Finally, it is now clear that the weakest topology on o ACI) that makes all of the functions x continuous is just the relative Gel'fand topology on A(A) - h(I), from which we conclude that A(I) and A(A) - h(l) are homeomorphic. This proves parts (i) through (iii). Next let ~: A - A/I denote the canonical homomorphism; that is, ~(x) = x + I, x E A. If T E ~(A/I), then it is evident that W =T 0 ~ is a complex homomorphism on A, where 0 denotes the usual composition of mappings. Moreover, if x E I, then

7. Regular Commutative Banach Algebras

176

w(x)

=T 0

q>(X)

= T(X

+

I)

= T(I) = 0,

whence we see that w E h(I). Conversely, if w E h(I), then we define T on A/I by setting T(X + I) = w(X) , x E A. This is well defined since, if x,y E A and x - y E I, then T[(X

+

I) - (y

+

I)] = T(X - Y + I)

= w(x

= 0,

- y)

as w E h(I). Thus T E 6(A/I), and w = T Q q>. In this way we obtain a bijective mapping from 6(A/I) to h(I). Furthermore, it is easily verified that, if

(x

+

I)

o

o

denotes ..

the Gel'fand transform of x + I E A/I, then (x + I) = xlh(I) and that 6(A/I) and h(I) are homeomorphic. This proves parts (iv) through (vi). o

Finally, A/I is semisimple if and only if (x + I) (T) = 0, Q T (6(A/I), implies x E I. But it is readily seen that (x + I) if and only if x E k[h(I)]. Since k[h(I)] ~ I, by Proposition 7.1.1(vii), we conclude that A/I is semisimple if and only if I=k[h(I)]. o Before we establish the theorem indicated at the beginning of this section we shall make a definition and prove some preliminary results. Definition 7.3.1. Let A be a commutative Banach algebra. We say that A is normal if, whenever K C 6(A) is compact, E C 6(A) is closed and K n E =~, there exists some x E A such that

..

(i)

X(T)

= 1,

T

(ii)

X(T)

•

= 0,

TEE.

E K.

Corollary 7.2.2 says precisely that I.I(G), G being a locally compact Abelian topological group, is normal. We shall see that the

=

°

7.3. Normal Algebras

177

same is true of every regular commutative Banach algebra. Lemma 7.3.1. Let A be a regular commutative Banach algebra. If T E ~(A), then there exist some x E A and some open set U C b(A) containing T such that •x(w) = 1, w E U.

• Proof. Clearly there exists some yEA such that yeT) ~ o. Let U be an open neighborhood of T with compact closure such that • yew) ~ 0, w € cl(U). Such a neighborhood exists, as ~(A) is locally compact. Since A is regular, we have cl(U) = h(k[cl(U))). Set I = k[cl(U)). Then I is a closed ideal, and from Theorem 7.3.1

=

o

•

we see that A(A/I) h(l) = cl(U) and (y + I) = yICl(U). In particular, (y + I) never vanishes on A(A/I), and ~(A/I) is compact. Moreover, A/I is semisimple, as k[h(I}) = k[cl(U)) = I. Consequently we may apply Corollary 6.2.2 to deduce that A/I has an identity. • Thus !here exists some x E A such that (x + I) o = xlh(I) that is, x(w) = 1, w E cl(U), which completes the proof.

= I, o

Corollary 7.3.1. Let A be a regular commutative Banach algebra. If K C ~(A) is compact, then there exists some x E A such • that X(T) = 1, T E K. Proof. From Lemma 7.3.1 we see that for each T E K there exist some xT E A and some open neighborhood UT of T such that •xT(w) = I, w E UTe Obviously the family {U I T E K) is an open T covering of K, and so there exist TI ,T 2, .•. ,Tn such that n K C Uk=IU Tk . A straightforward calculation then reveals that x = x 0 x 0 ••• 0 x in A is such that •X(T) = 1, T E K, Tl T2 Tn where x 0 y = x + y - xy. o It is perhaps worthwhile pointing out that this corollary, combined with Corollary 6.2.2, gives us another sufficient condition for a Banach algebra to have an identity.

178 ...-

7. Regular Commutative Banach Algebras

Corollary 7.3.2. Let A be a semisimple regular commutative Banach algebra. If A(A) is compact~ then A has an identity. This corollary, combined with Theorem 4.7.1, yields a partial converse to Corollary 4.7.5. Corollary 7.3.3. Let G be a locally compact Abelian topolo" is compact, then G is discrete. gical group. If G Proof. Since LI(G) is a semisimple regular commutative " we conclude from Corollary 7.3.2 Banach algebra and A(Ll(G)) = G, that LI(G) has an identity. But, by Theorem 4.7.1, this occurs if and only if G is discrete. o The normality of regular commutative Banach algebras is an immediate consequence of the next theorem. Theorem 7.3.2.

Let

A be a regular commutative Banach algebra.

If K C A(A) is compact, E C A(A) is closed, is any closed ideal in A for which h(l) = E, some x E I such that (i)

(ii)

" X(T)

= 1,

K n E =~,

and I then there exists

T E K•

• X(T) = 0, TEE.

Proof. Let I be a closed ideal such that E = h(I); for example, I could be keEl. By Theorem 7.3.1, A(I) is homeomorphic to A(A) - h(l) = A(A) - E. Since this latter set is open in A(A), it follows at once from the regularity of A that the hullkernel topology on A(I) is Hausdorff, and so, by Theorem 7.1.2, 1 is a regular commutative Banach algebra. Clearly K C A(A) - E = A(I) is compact, and therefore from Corollary 7.3.1 we deduce the existence of some x € I whose Gel'fand o transform x is identically one on K. But, by Theorem 7.3.1, o " " = 1, T € K. However, x = xlA(A) -h(l)' from which we see that X(T)

7.3. Normal Algebras since x E I, complete.

179

•

we have X(T)

Corollary 7.3.4. then A is normal.

= 0,

T

E h(l) = E, and the proof is

o If A is a regular commutative Banach algebra,

-

In the case that 6 (A) is compact it is immediate that the Gel'fand representation A of a regular commutative Banach algebra A even separates disjoint closed sets. This observation should help explain the origin of the term "normal" as applied to Banach algebras.

CHAPTER 8 IDEAL THEORY 8.0. Introduction. One of the more interesting portions of the theory of commutative Banach algebras is the study of the structure of closed ideals. As we shall see, any attempt to completely describe the closed ideals in an arbitrary commutative Banach algebra is hopeless, although the task may be relatively easy for certain specific algebras. Nevertheless a number of questions concerning closed ideals can be investigated fruitfully. The first such question we shall take up is to determine sufficient conditions under which a proper closed ideal I in a commutative Banach algebra A is contained in some maximal regular ideal. That is, when is it the case that h(l); ~ for each proper closed ideal I in A? Of course, as we saw previously, if A has an identity, then this is certainly the case, and similarly every proper regular closed ideal is always contained in a maximal regular ideal. However, in algebras without identity it is not entirely clear when the desired phenomenon occurs. We shall prove in the first section that, if A is a semisimple regular commutative Banach algebra such that the elements of A whose Gel'fand transforms have compact support are norm dense in A, then h(l) ~ ~ for each proper closed ideal I in A. We shall see that one important example of such a Banach algebra is LI(G), where G is a locally compact Abelian topological group. This result has a number of interesting consequences. For example, we shall show that the linear subspace of LI(G) spanned by the translates of some f E LI(G) is dense in LI(G) if and only if 180

8.1. Tauberian Commutative Banach Algebras

181

A

f never vanishes. Moreover, the indicated result will be used to establish some Tauberian theorems. We shall prove two such theorems, one due to Wiener and one to Pitt. The second question about ideals that we shall consider at some length is the determination of conditions under which the hull of a proper closed ideal I uniquely determines I. Since we know that in a regular commutative Banach algebra h(I) is closed and h(k[h(I)]) = h(l), we see that I and k[h(I)] are two closed ideals with the same hull, and we shall see that the question of when h(l) determines I is equivalent to determining when I = k[h(I)]. In other words, the problem will be to determine when a proper closed ideal I is the intersection of the maximal regular ideals containing I. We shall refer to this as the problem of spectral synthesis. Although it is, in general, a very difficult problem, some general results are available. For instance we shall see that, if A is a semisimple regular commutative Banach algebra that satisfies certain additional requirements known as Ditkin's condition, then I = k[h(I)] for each proper closed ideal I in A such that the topological boundary of h(l) contains no nonempty perfect set. The proof of this result, which appears in Section 8.5, is rather intricate and involves a number of results of independent interest. Although the problem of spectral synthesis is, in general, difficult, its solution for certain algebras is quite elementary. We shall, for instance, solve the problem completely for C(X), X being a compact Hausdorff topological space, and for Ll(G), G being a compact Abelian topological group. The problem for Ll(G) when G is noncompact is, however, exceedingly complex. In the final section of this chapter we shall briefly mention some other questions and results concerning closed ideals. 8.1. Tauberian Commutative Banach Algebras. We have already observed that, if A is a commutative Banach algebra, then every proper regular ideal in A is contained is some maximal regular

182

8. Ideal Theory

ideal. However, it need not generally be the case, when A is without identity, that every proper closed ideal is contained in a r-aximal regular ideal. The main result of this section will be to show that this does occur whenever A is a semisimple regular commutative Banach algebra such that {x I x E A, x• E C (A(A))) is norm dense c in A. In particular, we shall see that, if G is a locally compact Abelian topological group, then every proper closed ideal in Ll(G) is contained in some maximal regular ideal. Some further applications of these results will be discussed in the next section. To begin we make some definitions and some elementary observations. Definition 8.1.1. Let A be a commutative Banach algebra and suppose E C A(A) is closed. Then I (E) will denote the set of • 0 all x E A for which x vanishes identically on some open set Ox C A(A) such that Ox ~ E; and Jo(E) will denote the set of • € Cc(A(A)). x E IoCE) such that x Given a closed set E C A(A), it is evident that J o (E) C I 0 (E), that Jo(E) and I o (E) are ideals in A, and that cl[Jo(E)) = cl[Io(E)). Moreover, it follows from the definition of k[h(I)) that, if I is any closed ideal such that h(l) = E, then k[h(I)) ~ Cl[Io(E)). If A is semisimple and regular, then cl[Io(E)] is actually the smallest closed ideal in A whose hull is E. This is the content of the second part of the next theorem. Theorem 8.1.1. Let A be a regular commutative Banach algebra and let E C 6(A) be closed. Then (i)

h(cl[I o (E)])

= h(cl[Jo(E)]) = E.

(ii) If A is semisimple, then cl[Io(E)] is a closed ideal such that h(l) = E.

C

I

whenever

I

C

Proof. It is apparent that E C h(cl[Jo(E)]). On the other hand, if T ~ E, then let U be an open neighborhood of T with

A

8.1. Tauberian Commutative Banach Algebras

183

compact closure such that cl(U) n E = ,. Since A is regular, there exists some x E A such that •X(T) ~ 0 and •x(w) = 0, w E ~(A) - U. Thus the support of •x lies in cl(U) and so is • compact, and x vanishes identically on the open set

ox

= ~(A)

- cl(U)

which contains E. Hence x E J eE) C cl[1 (E)], and •X(T) o 0 Consequently T ~ h(cl[Jo(E)]), and so E = h(cl[Io(E)]).

~

o.

To prove part (ii) of the theorem, suppose I C A is any closed ideal such that hel) = E. If x E Jo(E), then •X E C (~(A)), and • c there exists an open set 0 ~ E on which x vanishes identically. x • If K denotes the compact support of x, we see at once that K n E = ,. Thus, by Theorem 7.3.2, there exists some y E I such that •yeT) = I, T E K, and •yeT) = 0, TEE. However, it is easily • • verified that X(T) =• X(T)y(T). T E ~(A), whence, since A is semisimple, we conclude that x = xy E I.

I

Therefore Jo(E) is closed.

C

I,

and so cl[Io(E)]

= cl[Jo(E)]

C

I,

as

o

The result indicated in the introduction is a simple corollary of this theorem. Definition 8.1.2. Let A be a commutative Banach algebra. Then A is said to be Tauberian if (x l x E A, ~ E C (~(A))l is c norm dense in A. Corollary 8.1.1. Let A be a Tauberian semisimple regular commutative Banach algebra. If I C A is a proper closed ideal, then there exists some maximal regular ideal MeA such that M ~ I.

Proof. Suppose I is a closed ideal that is contained in no maximal regular ideal; that is, E = h(I) = ,. Then clearly Jo(E) = (x I x E A. x• E Cc(~(A»], and so, by Theorem 8.1.1, we have A = cl[Jo{E)] C I, as A is Tauberian. Hence I is not proper. 0

8. Ideal Theory

184

Since all ideals in an algebra with identity are regular, Corollary 8.1.1 has nontrivial content only in the case of algebras without identity. An important collection of such algebras that are Tauberian are the algebras Ll(G). Theorem 8.1.2. group. Then Ll(G) Banach algebra.

Let G be a locally compact Abelian topological is a Tauberian semisimple regular commutative

Proof. We need only prove that Ll(G) is Tauberian, as the other assertions were established in Theorem 4.7.4 and Corollary 7.2.3. Appealing to Plancherel's Theorem (Theorem 7.2.1), we see at once that the set of f E L2 (G) whose Plancherel transform •f • is equal almost everywhere to some element of C (G) is a norm-dense c linear subspace of L2 (G). We denote this subspace by L~(G). Moreover, if g,h E L2 (G), then g * h E C (G), whence we deduce that • • • c • gh E Ll(G) n L2 (G) and (gh) = g * h. Note that (gh) here is actually the Fourier transform of gh. Furthermore, each f ( Ll(G) can be written as f = f l f 2• where fk E L2 (G) , k = 1,2, from which it follows easily that (gh I g,h € L~(G)] is norm dense in L1(G), since L~(G) is norm dense in L2 (G).

c··

However, (f I f E LI(G), whence we conclude that LI(G)

·

f € Cc(G)) ~

(gh is Tauberian.

I

g,h € L~(G)J, 0

Corollary 8.1.2. Let G be a locally compact Abelian topological group. If I C Ll(G) is a proper closed ideal, then there exists some maximal regular ideal Me LI(G) such that M ~ I.

as

Of course, if G is discrete. then the corollary is trivial, Ll(G) has an identity.

Recalling the definition of the translation operators Ts defined in Definition 4.7.1, we can apply Corollary 8.1.2 to obtain the following interesting theorem:

8.1. Tauberian Commutative Banach Algebras

185

Theorem 8.1.3. Let G be a locally compact Abelian topological group, let f E LI(G), and suppose I denotes the closed linear subspace of LI(G) spanned by {Ts(f) I s E G). Then the following are equivalent:

(ii)

•

~

fey)

~

0,

•

E G.

Proof. We claim first that I is a closed ideal in To see this it suffices to show that g * f £ I for each since g * Ts(f) = Ts(g * f), s E G, and I is invariant the translation operators T , s E G. To this end we note s ~ ( L (G) is such that CD

IG

h(t)~(-t) dArt)

=0

LI(G). g £ L1 (G) under that, if

(h E I),

then f * ~(s)

= IG

f(s - t),(t) dA(t)

= IG

f(s

=

IG

+ t)~(-t)

dArt)

T_s(f)(t)~(-t) dArt)

=0 that is,

f *

~

= O.

iG g *

(s

Hence for any g € LI(G)

f(t)~(-t) dArt)

= (g *

E G),

we have

f) * ~(O)

= g * (f *

~)(O)

= o. Thus, since the dual space [OSI' pp. 289 and 290; El , of the Hahn-Banach Theorem each g E LI(G). Hence I

of pp. [L, is

LI(G)

can be identified with Lm(G) 215-220, 239 and 240], a consequence p. 90] shows that g * f £ I for a closed ideal in L1 (G).

8. Ideal Theory

186

Now if I ~ Ll(G), then I is a proper closed ideal in Ll(G), whence by Corollary 8.1.2 there exists some maximal regular ideal Me L1(G) such that M~ I. Thus from Theorem 4.7.3 we see that A • there exists some y E G such that M = {g I g E Ll(G), g(y) = 0). A In particular, we would have fey) = 0, contradictiong the assumption that •f never vanishes. Thus part Cii) of the theorem implies part (i).

• Conversely, suppose fCY) = 0 for some y E G. An elementary A • A computation reveals that T (f) (w) = (-s,w)f(w), s E G and w E G, s. whence we deduce that T (f) (y) = 0, s E G. It is then obvious A s that hey) = 0, h E I, and so I is contained in the maximal regular A ideal M = {g I g E L1CG), g(y) = oj. Hence I is proper, and so part Ci) implies part (ii). o A

The fact established in the preceding proof that the closed linear subspace I spanned by {Ts(f) I s (G) is a closed ideal is a special case of a generally valid result in Ll(G): a closed linear subspace I of LleG) is an ideal if and only if it is translation invariant; that is, if and only if Ts(g) E I, s E G, whenever gEl. We state this result as the next theorem leaving the proof to the reader. Theorem 8.1.4. Let G be a locally compact Abelian topological group and let I C Ll(G). Then the following are equivalent: (i)

I

is a closed ideal.

Cii)

I

is a closed translation-invariant linear subspace.

This is also an appropriate point to mention some approximation results for LI(G) that are easy corollaries of the fact that Ll(G) is Tauberian. Corollary 8.1.3. Let G be a locally compact Abelian topological group. If f E L1(G) and & > 0, then there exists some A A V E LI(G) such that v E Cc(G) and IIf - f * vIII < &.

8.1. Tauberian Commutative Banach Algebras

187

Proof. From Theorem 4.7.2 we know that LlCG) contains an approximate identity, and so there exists some u E LICG) such that IIf - f * ull l < 1/2. Then, s!nce LICG) is Tauberian, there exists some v E LI CG) such that v ( CC C~) and lIu - vIII < 1/2l1f Il 1 • Note that we may assume that f ~ 0 since the result is trivially valid in this case. Hence IIf - f * vIII ~ IIf - f * ull l

+

IIf * u - f * vIII

~ IIf - f * ull l

+

IIflllliu - vIII

£

£

2

2

<-+-

= I.

o

Corollary 8.1.4. Let G be a compact Abelian topological group. If f E LICG) and I > 0, then there exists a finite linear combination of continuous characters of G, say Ey EG ayC· ,y), ay E~, where only finitely many ay are nonzero, such that IIf - f * [~EG ayc·,y)]U I < c. Proof. 8y Corollary 8.1.3 there exists some v E LlCG) such • • • that v £. Cc (G) and IIf - f * vIII < £. Since G is discrete by Corollary 4.7.5, the support of must be finite, say Yl'Y2' ••• 'y • •

v•

n

It is then evident that v = Ik=lvCYk)X(Yk)' where X(Yk) denotes the characteristic function of (Yk), whence, from the semisimplicity of Ll(G) (Theorem 4.7.4) and Corollary 4.7.4, we conclude that

•

v = ~=lvCYk)(·'Yk)·

0

A finite linear combination of continuous characters of a locally compact Abelian topological group G is usually called a trigonometric polynomial. The origin of the terminology is clear on considering the group G = f. Utilizing these corollaries it is not difficult to prove the following result, the details being left to the reader:

8. Ideal Theory

188

Corollary 8.1.5. Let G be a locally compact Abelian topological group. Then LI(G) contains an approximate identity (u a ) such that (~) c C (~). Moreover, if G is compact, then LI(G) Ot c contains an approximate identity consisting of trigonometric polynomials. Theorem 8.1.2 and the idea of the proof of Corollary 8.1.3 also provide us with the following useful result: Corollary 8.1.6. Let G be a locally compact Abelian topolo• gical group. If KeG is compact and e > 0, then there exists some f E LI(G) such that (i)

(ii)

•f E C (G). • c

..

= 1,

fey)

Y E K•

•

Proof. If G is compact, then from Corollary 7.3.3 we see that G is discrete, and so Ll(G) has an identity e. Clearly

..~(y)

.

and lIeU I < 1 + e. On the other han~, suppose G is noncompact. Then there exists an open set WC G such that • - W, then E is closed W~ K, and cl(W) is compact. If E = G and K n E whence, by Theorem 7.3.2 or Corollary 7.3.4, we • deduce the existence of some h £ Ll(G) such that hey) = 1, Y E K,

= I,

.

Y E G,

="

and hey) = 0, y E E. In particular, be such that 6 < min(cllhlll,e).

..

.

h E Cc(G).

6 >

Now let

°

From the proof of Theorem 4.7.2 we see that there exists some u C LI(G) such that lIulil = 1 and IIh - h * ull l < 6/3, and, since Ll(G) is Tauberian (Theorem 8.1.2), there exists some g E Ll(G)

•

•

such that g E C (G) and !lu - gli l < 6/3I1hIl1. c .. • Evidently f E LI(G), f E Ce(G), and

- .

fey) = hey) = 1

+

+

Set

f

=h

+

g - h * g.

. ..

g(y) - h(y)g(y)

• g(y) - •gCy) = 1

(y E K).

8.2. Two Tauberian Theorems

189

Moreover,

< IIglll

+

IIh - h * gill

< IIglll

+

IIh - h * ull l

< lI u ll l

+

< I

6

6

+ -

311 h l;1 6

6

3 311hlll e- + e- + e < 1 +3 33

=1

+

+

+

3

+-

+

IIh * u - h * gill IIhlllliu - gill

6

+3

e.

c

8.2. Two Tauberian Theorems. The theorem of Tauber, from which the name "Tauberian theorem" originates, is the following: Suppose (ak ) is a sequence of complex Rumbers such that the power series

~=O ak,k converges in (,

1,

E

Q;,

1'1

< I).

If

limk_CDkak = 0

and lim Ek=O akrk = a, r-I O
8. Ideal Theory

190

For the sake of completeness we make one definition before giving Wiener's theorem. Definition 8.2.1.

Let

gical group and suppose G. Then limt_mg(t) set KeG such that

G be a locally compact Abelian topolo-

g is a complex-valued function defined on

=a

if, given ~ > 0, there exists a compact 19(t) - al < ~, t E G - K. A function g is

> 0, there exist a comand a compact neighborhood U of 0 in G such

said to be slowly oscillating if, given pact set that

KeG

(g(t) - g(s)l <~, t - s € U and

~

t E G - K.

Obviously every uniformly continuous function on G is slowly oscillating. Theorem 8.2.1 (Wiener's Tauberian Theorem). Let compact Abelian topological group and let g E Lm(G). h E LI(G)

exist some

,..

(i)

(ii) Then

hey)

..

a E C such that

0, y E G.

lim t -t»g * h(t)

limt_cx$ * f(t) Proof.

g

~

and some

G be a locally Suppose there

= aJG

= aJG

h(u) dA.(u).

feu) dA,(u)

for each

f

in

LI(G).

We remark first that the limits are meaningful since

* f, fELl (G),

tion 4.7.2.

is a bounded continuous function on G by ProposiDenote by I the set of all f E Ll(G) such that

lim t -t»g * f(t) = aJG feu) dl.(u). Clearly I is a closed linear subspace of LI (G), and I ~ (OJ because h E 1. Moreover, I is translation invariant since, if f E I, then lim g * T (f) (t) t-t» S

=

lim T (g * f) (t) t-m s

=

lim g * f (t - s) t-t»

= lim g * f(t) t-m

8.2. Two Tauberian Theorems

191

= aiG

feu) dA.(u) (s E G) •

= aiG Ts(f)(u) dA.(u)

Thus, in particular,

I

contains the closed linear subspace of

•

•

LI(G) spanned by {Ts(h) I s E G). Since hey) ~ 0, y E G, it follows immediately from Theorem 8.1.3 that I = LI(G). This completes the proof. o Wiener's original result was, of course, only for G = R, and his proof is quite intricate. It was in proving this theorem that Wiener needed the result about absolutely convergent Fourier series, now generally known as Wiener's Theorem (Theorem 4.6.2), which we have already discussed. Wiener's proof is available in [Wr l , pp. 7297] (see also [Pi, pp. 43-92]). We note that Corollaries 8.1.1 and 8.1.2 and Theorem 8.1.3 are also referred to as Wiener's Tauberian Theorem. It may not be completely clear why Wiener's Tauberian Theorem carries such a name, and we shall not pursue the reasons in any detail. The following theorem should, however, provide some support for the terminology: Theorem 8.2.2 (Pitt). Let G be a locally compact Abelian topological group and let g be a bounded, measurable, slowly oscillating function on G. Suppose there exist some hELl (G) and some a E C such that (i)

(ii)

•hey)

~

0, y

• E G.

limt _~ * h(t)

= aiG

h(u) dA.(u).

Proof. Considering g as an element of once from Wiener's Tauberian Theorem that

LCII (G),

we see at

8. Ideal Theory

192

lim g • f{t)

= aiG

feu) dA. (u)

t .. CD

Now, given £ > 0, since g is slowly oscillating, there exist a compact set KI C G and a compact neighborhood U of 0 € G such that Ig(u) - g(v)1 < e/2, u - v E U and u E G - KI . Consequently, if fo

= Xu/A.(U),

then

Ig(t) - g * fo (t) I

fo (L1(G) -

rlur JU get - s) dA(s) I

Iu

19(t) - get - s) I dA.(s)

= 19(t) ~ 1(~)

and

e

(t E G - KI ).

<2 But since lim g * fo (t) = aIG fo (u) dA.(u)

t-m

= a, there exists some compact set

.CG

K?

such that

Therefore, if K = Kl U K2 , we see that t E G - K, which proves the theorem.

1get) - al <

i,

o

8.3. The Problem of Spectral Synthesis. Given a commutative Banach algebra A, it would obviously be of considerable interest to be able to describe all the closed ideals in A. In general this is apparently a hopeless task, although we shall see shortly that for some specific Banach algebras rather simple descriptions of the closed ideals may be available. Some progress can be made on the general question by considering the related problem of determining when a closed subset of the maximal ideal space 6(A) can be the hull of more that one closed ideal in A. In this and the following sections we shall concentrate our attention mainly on this question.

8.3. The Problem of Spectral Synthesis

193

First, let us see what relation this sort of problem has to the description of closed ideals. Let A be a semisimple regular commutative Banach algebra and suppose E C 6{A) is closed. Then from Theorem 8.1.1 we see that

E

= h(cl[1 o (E)]) = h(cl[J0 (E)])

and that

cl[lo(E)] is the smallest closed ideal in A whose hull is E. We recall that I (E) consists of all the x ~A such that x• vanishes o identically on some open set o C A(A) such that 0 ~ E. x x

On the other hand, from the definition of the hull-kernel topology and Proposition 7.1.1 we see that keEl is a closed ideal in A such that h[k(E)] = E, and if I C A is a closed ideal for which h(l) = E, then I C keEl; that is, keEl is the largest closed ideal whose hull is E. Thus, if there were precisely one closed ideal

I

such that

h(l) = E, then we would have to have 1= cl[I (E)] = keEl o In particular, I would consist of those x E A such that T € h(l)

=• k[h(I)]. X(T) = 0,

= E.

We summarize these observations in the following proposition: Proposition 8.3.1. Let A be a semisimple regular commutative Banach algebra. If E C A(A) is closed, then the following are equivalent: (i)

cl[Io(E)]

(ii) If I h(J) = E, then (iii) If I = k[h(I»).

I

= keEl.

and J I = J.

are closed ideals in

is a closed ideal in

A such that

A such that

h(l)

h(l)

= E,

=

then

However, it may very well be the case that, given a closed set E in 6(A), there exist distinct closed ideals I and J such that h(l) = h(J) = E, equivalently, so that CI[lo(E)] ~ keEl. We shall mention an example of this in LI QR3) following Theorem 8.3.4.

8. Ideal Theory

194

With this in mind we make the following definition: Definition 8.3.1. Let A be a semisimple regular commutative Banach algebra. Then a closed set E in ~(A) is said to be a set of spectral synthesis if I = keEl is the only closed ideal in A such that h(l) = E. Proposition 8.3.1 provides several equivalent formulations of this definition. In particular we note that, if E is a set of spectral synthesis for A and I is a closed ideal such that h(I) = E, then I = k[h(I)]; that is, I is the intersection of all the maximal regular ideals in A containing I. The problem of determining which closed sets in A(A) are sets of spectral synthesis is called the problem of spectral synthesis. The development of Section 8.1 provides us with a necessary and sufficient condition that the empty set be a set of spectral synthesis. Theorem 8.3.1. Let A be a semisimple regular commutative Banach algebra. Then the following are equivalent: (i) (ii)

A is Tauberian. ~

is a set of spectral synthesis.

Proof. If E = ~, then it is immediately apparent that J (E) 0 • is precisely the set of all x €A such that x € C (~(A)). But c A is Tauberian if and only if cl[J (E)] = cl[I (E)] = A, whence o 0 we conclude, since keEl = k(~) = A, that E = ~ is a set of spectral synthesis if and only if A is Tauberian. o In general the problem of spectral synthesis is a complicated one. Before we pursue it further in the abstract we wish to consider several specific examples. First, we shall describe completely the closed ideals in C(X), X being a compact Hausdorff topological

8.3. The Problem of Spectral Synthesis

195

space. An immediate corollary of this result will be that every closed subset of X is a set of spectral synthesis for C(X). Theorem 8.3.2. Let X be a compact Hausdorff topological space and suppose I c CeX). Then the following are equivalent: (i)

I

is a closed ideal in C(X).

(ii) There exists a closed set E C X such that {f I f £ C(X), f(t) = 0, t E EJ.

I = keEl =

Proof. Obviously part (ii) of the theorem implies part (i), so suppose that I is a closed ideal in C(X). Ultimately we shall wish to apply the Stone-Weierstrass Theorem [L, p. 332] to I, and with this is mind we shall first show that I is closed under complex conjugation. Indeed, let f £ I and suppose s > O. Then define gs ~ C(X) by gs(t) = f(t) [f(t)]2/(S + If(t)1 2), t E X. Clearly gs E I, as I is an ideal, and

(t

E X).

Hence clf(t)1 c + If(t)1 2

<./s

-2

The validity of the last estimate is easily verified.

(t E X).

8. Ideal Theory

196

< ./£/2, from which it folConsequently we see that lig t: - £11CX)lows that f E I, as I is a closed ideal. Thus I is closed under complex conjugation.

Now let E = h(I). From Theorem 7.3.1 we see that I is a commutative Banach algebra whose maximal ideal space a(I) is homeomorphic to X - h(I) = X - E and that the Gel'fand transform of f, as an element of the Banach algebra I, is just the Gel'fand transform of f, as an element of C(X), restricted to X - E; that is, the Geltfand transform of f E I is just the restriction of f to X-E. Thus, since X - E is open and each f in I vanishes identically on E = h(I), we see that, if f E I, then f is a continuous function on X vanishing identically on E. It is apparent that the set of all such functions in C(X) can be identified with C (X - E). Hence the proof will be complete if we can show o that I, considered as a subalgebra of C (X - E), is actually all o of Co(X - E). Evidently I is a closed subalgebra of C (X - E), which is o closed under complex conjugation. Moreover, I separates the points of X - E because, if t,s E X - E, t ~ s, then, since C(X) is regular, there exists some g E C(X) such that get) = 1, g(s) = o. Since t ~ E = h(I), there exists some f E I such that f(t) ~ O. Then fg E I, fg(t) ~ 0, and fg(s) = 0, and so I separates points. Finally, there exists no t € X - E such that f(t) = 0, f E I, because such a t would have to be in h(l) = E. Thus all the hypotheses of the Stone-Weierstrass Theorem [L, p. 332] are fulfilled, and so I = C (X - E) = (f I f E C(X), f(t) = 0, t E EJ. o

This completes the proof.

c

Corollary 8.3.1. Let X be a compact Hausdorff topological space and suppose E C X. Then the following are equivalent: (i) (ii)

E is closed. E is a set of spectral synthesis for

C(X).

8.3. The Problem of Spectral Synthesis

197

Moreover, every proper closed ideal I in C(X) is the intersection of the maximal ideals containing I·, that is, I = k[h(I)]. Proof. If E is closed and I is a closed ideal in C(X) such that h(l) = E, then, by Theorem 8.3.2 and the regularity of C(X), we see that there exists some closed set F C X such that I = kef) and E = h(l) = h[k(F)] = F, that is, I = keEl. Thus E is a set of spectral synthesis for C(X). o The analog of Theorem 8.3.2 is also valid for LI(G) provided G is a compact Abelian topological group. Recall that, since G • is discrete (Corollary 4.7.5), and is compact, the dual group G • is closed. hence every subset of G Theorem 8.3.3. Let G be a compact Abelian topological group and suppose Ie LI(G). Then the following are equivalent: (i)

I

is a closed ideal in

LI(G) •

• such that (ii) There exists a set E C G (f I f (LI(G), •fey) = 0, y E EJ.

I

= keEl =

Proof.

As before, we need only prove that part (i) implies part (ii). So let I be a closed ideal and let E = h(l). Then from Proposition 7.1.I(vii) we see that I C k[h(I)] = keEl.

•

To establish the reverse containment we note that, if y E G - E, • then (.,y) E I. This follows from the fact that, if y E G - E = • - h(I), then there exists some gEl such that g(y) • G = 1. Since G is compact, we see that (.,y) £ L1 (G) and g * (·,y)(t)

= IG

(t - s,y)g(s) dA(s)

• = (t,y)g(y)

= (t,Y) Thus

(.,y) E I,

as

I

is an ideal.

(t E G).

198

8. Ideal Theory

Consequently I contains all the trigonometric polynomials of the form 1:y (G _ E ay (. ;v) . Hence if f ( k [h (1)], then we see that, for any trigonometric polynomial 1:y E ay (. , y), JI

a

=

.fey) = 0, y E E

=

•

1:.a fey) (t,y) yEG Y ~

•

(t ( G),

a"f(y) (t,y)

yEG-E

since = h(I) = h(k [h(l)]). Thus f * [1:y E aay (. ,y)] belongs to I for each f € k[h(I)] and each trigonometric polynomial. But then, appealing to either Corollary 8.1.4 or 8.1.5, we conclude that f E I, as I is closed, that is, k[h(I)] C I. Therefore

I

= k[h(I)] = k(E), and part (i) implies part (ii).O

Corollary 8.3.2.

•

Let G be a compact Abelian topological group .

If E C G, then E is a set of spectral synthesis for Ll(G). Moreover, every proper closed ideal I in LI(G) is the intersection of the maximal regular ideals containing Ii that is, I = k[h(I)]. An examination of the proof of Theorem 8.3.3 reveals that, if

I is a closed ideal in LI(G), G being a compact Abelian topological group, then every f E I is the limit in LI(G) of trigono-

•

metric polynomials of the form EYEG-h(I) ayf(Y)C"Y) a~d that all such polynomials belong to I. Thus, since oCf) = Ref) or • u (oj, fELl (G), depending on whether G is finite oCf) = Ref) or infinite, we see that the elements of I are determined by those • such that y E• G such • that fCY) ~ 0; that is, by those y E G •fCy) E oCf) and fCy) • ~ O. Hence, loosely speaking, we see that an element fEr can be recovered or synthesized from a knowledge of the nonzero numbers in oCf). These remarks should supply some insight into the origin of the term "spectral synthesis".

8.3. The Problem of Spectral Synthesis

199

This basis for the terminology is even more apparent if one considers the Banach algebra L2 (G) in place of Ll(G), G being a compact Abelian topological group, since then one has all the machinery of Hilbert space theory available. For a related, but somewhat different, motivation the reader is referred to [Ru l , pp. 183-186]. If G is a noncompact locally compact Abelian topological group, then the structure of the closed ideals of LICG) is vastly more complicated than in the case of compact groups, and it is no • is a set of spectral longer the case that every closed subset of G synthesis. Moreover, in this case there exist no known necessary • and sufficient conditions for a closed set E C G to be a set of spectral synthesis. Since it would entail a considerable digression to investigate in any detail the problem of spectral synthesis in LICG), we shall content ourselves here with some observations on the problem without any proofs. Some additional results will be proved in Section 8.6. First we note that the failure of spectral synthesis in LICG), G being a noncompact locally compact Abelian topological group, is universal; that is, it occurs for all such groups. This is a celebrated result of Malliavin, which we state as the next theorem. Theorem 8.3.4 CMalliavin). Let G be a noncompact locally compact Abelian topological group. Then there exists some closed • set E C G that is not a set of spectral synthesis for LICG). Prior to Malliavin's theorem it was known that certain groups contained closed sets that were not sets of spectral synthesis. For example, L. Schwartz showed that E = {Ct l ,t 2,t 3) I t~ + t~ + t~ = 11 is such a set in JR3. The analogous assertion is actually valid in ~, n ~ 3, but false in Rand Ff. On the other hand, sets that fail to be sets of spectral synthesis can be fairly complicated . • For instance, if G is noncompact and 0 eGis open and nonempty, then 0 contains a closed subset E that is not a set of spectral

8. Ideal Theory

200

synthesis and such that

E is homeomorphic to the Cantor ternary

set. Nevertheless the Cantor ternary set itself is a set of spectral synthesis in Ll (lR) • Simple operations with sets of spectral synthesis may not produce sets of spectral synthesis. For example, the intersection of sets of spectral synthesis may not be such a set. Indeed, the sets

and

E2 = {(t l ,t 2,t 3)

I t~

are both sets of spectral synthesis in

+

t~

+

t; ~ I}

Ll~)' but, as we have

noted, E = El n E2 is not. On the other hand, the union of two disjoint sets of spectral synthesis is again a set of spectral synthesis. It is unknown whether the same is true for nondisjoint sets of spectral synthesis. More complete discussions of the problem of spectral synthesis in Ll(G) can be found, for instance, in [HR2 , pp. 484-605; Ru l , pp. 157-191]. We now wish to return to the problem in general semisimple regular commutative Banach algebras. 8.4. Local Membership in Ideals. In the next section we shall prove Ditkin's Theorem, which provides the most general known sufficient condition for a closed set to be a set of spectral synthesis. First, however, we need some preliminary results which supply sufficient conditions for an element x in a semisimple regular commutative Banach algebra A to belong to a given ideal I in A. The role of these results in the problem of spectral synthesis will become apparent in the next section. We begin with some definitions.

8.4. Local Membership in Ideals

201

Definition 8.4.1. Let A be a commutative Banach algebra and let I be an ideal in A. An element x (A is said to belong locally to I at the point T € a(A) if there exist an open neigh• borhood U of T and some y € I such that x(w) = •yew), w E U. If a(A) is noncompact, then x is said to belong locally to I at infinity if there exist a compact set K c a(A) and some y E I

•x(w)

such that

=

•

y(w)~

w ( a(A) - K.

The theorem of this section asserts that, if A is a semisimple regular commutative Banach algebra and x € A belongs locally to an ideal I at every point in a(A) and at infinity when a (A) is noncompact, then x belongs to I. Before proving this result we need a lemma about continuous functions on normal topological spaces.

AC

Lemma 8.4.1. Let X be a normal topological space and suppose C(X) is a subalgebra such that

Ca)

A

contains the constant functions.

(b) If El ,E 2 are disjoint closed subsets of X, then there exists some f E A such that f(t) = 1, t (E I , and f(t) = 0, t

€ E2 •

If E C X is closed and E C ~=lUk' where Uk C X is open, k : 1,2, ••• ,n, then there exist hk € A, k : 1,2, •.• ,n, such that

(ii)

~(t) =

Proof.

0,

t

E X - Uk' k

= 1,2, ... ,n.

The lemma is evidently valid for

n

=1

on setting

El = E, E2 = X - Ul and applying the hypotheses of the lemma. If n = 2, then E C UI U U2 , and so E n (X - U2) C Ul is closed. Since X is a normal topological space, there exists an open set U C X such that E n eX - U2) cue cl(U) cUI (see, for example, [W2 ' p. 49]). Then cl(U) and X - U1 are disjoint closed subsets of X, and so there exists some hI € A such that hI (t) = I,

8. Ideal Theory

202

t E cl(U), and hl(t) = 0, t E X - Ul • Moreover, it is apparent that E and X - (U U U2) are also disjoint closed subsets of X, whence there exists some f E A such that f(t) = I, tEE, and f(t) = 0, t £ X - (U U U2). Set h2 = f - fh 1 • Clearly h2 E A, and if tEE, then

= 1, as f(t) = 1, t £ E. Furthermore, hl(t) = 0, t E X - U1 ' by the choice of hI' whereas if t E X - U2 ' then either t E (X - U2) nu or t E X - (U U U2). In the first case t E cl(U), and so h 2 (t) = f(t) - f(t)hl(t) = f(t) - f(t) as

= 0,

hl(t) = 1, t E cl(U).

In the second case h2 (t) = 0, as f(t) = 0, t E X - (U U U2). Thus hk(t) = 0, t E X - Uk' k = 1,2. Now suppose the conclusion of the lemma is valid for

n - 1,

where n > 3, and suppose E C ~=IUk. Then F = E n (X - Un) is closed and F C ~:~Uk. Again appealing to the normality of X, we deduce the existence of an open set U such that n-I Feu

U Uk" k=l Then, by the induction hypothesis, there exist C

cl(U)

C

gk E A, k = 1,2, ... ,n-I,

~-I such that ~=Igk(t) = 1, t E cl(U), and gk(t) = 0, t ( X - Uk' k = 1,2, •.• ,n - 1. Moreover, E C U U U, and so, applying the n lemma with two open sets, we see that there exist f and h in n A such that f(t) + h (t) = I, t (E, and f(t) = 0, t E X - U, n and hn(t) = 0, t ( X - Un. Hence on setting hk = fg k , k = 1,2, •.. ,n-l it is easily verified that tk=lhk(t) = 1, tEE, and ~(t) = 0, t E X - Uk' k = 1,2, •.. ,n.

This completes the proof of the lemma.

o

8.4. Local Membership in Ideals

203

The reader should observe that the lemma is actually valid without any assumptions of continuity on the functions in A. However, our applications of the lemma will only be in the case where A consists of continuous functions. Theorem 8.4.1. Let A be a semisimple regular commutative Banach algebra and suppose I C A is an ideal. If x E A belongs locally to I at every point of A(A) and at infinity when A(A) is noncompact, then x E I. Proof. We claim that, without loss of generality, we may assume that A has an identity. If this were not so, then I would still be an ideal in A[e] and x would belong locally to I at all points of A(A[e]) = A{A) U {TCDJ. Indeed, it is obvious that x belongs locally to I at each point of A(A), and x belongs locally to I at T since x belongs locally to I at infinity • CD and Y(T) = 0, yEA. Thus if the theorem is valid for algebras CD with identity, we conclude that x E I. Consequently we may assume that A has an identity. Then A(A) is a compact Hausdorff, and so normal, topological space, and A is a normal Banach algebra by Corollary 7.3.3. Thus • C C(A(A)). we may apply Lemma 8.4.1 to A Now x belongs locally to I at each point in A(A). Hence, given T E A(A), there exist an open neighborhood U of T and •• T some YT E I such that x{w) = YT(w), w E UTe Clearly the sets (U T I T E A{A)) form an open covering of A(A), and so there exists a finite number of the UT, call them Ul ,U 2 , .•. ,Un , such that A(A) = Uk=IU k • For the sake of notational simplicity we denote the YT corresponding to Uk by Yk , k = 1,2, ... ,n. Applying Lemma 8.4.1 to the closed set E = A(A), we deduce the existence of xk E A, k = 1,2, •.. ,n, such that ~=I;k{T) = 1, T E A(A), and ~k(T) = 0, T (~(A) - Uk' k = 1,2, •.. ,n. Evidently ~=lxkYk E I. Moreover, we claim that

8. Ideal Theory

204

" (1')

X

Indeed. let

(1' (

~(A)).

~(A)

and let Uk ,Uk •.•. ,U k denote those " ..1 2 m . Uk such that l' E Uk. Now Yk.(w) = x(w). W (Uk.' J = 1,2, ... ,m, J J so in particular these equations hold for W = 1'. On the other hand, l' (

x" k (1') = 0 if k ~ kj , j = 1,2, ... ,m, Hence we see that

= x" (1')

" since xk(w) = 0,

W

E

~(A)

- Uk.

n "

1: x k . (1')

j=l

J

" = x(1'). Therefore, since A is semisimple, we conclude that x = ~=IXkYk belongs to 1.

o

8.5. Ditkin's Theorem. We are now almost in a position to prove the theorem alluded to at the beginning of the preceding section. To describe the relation between Ditkin's Theorem and spectral synthesis we first need to make another definition. Definition 8.5.1. Let A be a commutative Banach algebra. Then A is said to satisfy Ditkin's condition at l' in ~(A) if, whenever x E A and l' E Z(x), there exist a sequence (x k ) C A and open neighborhoods Uk of T such that (i) (ii)

li~lIxxk -

xII

=

o.

" xk(w) = 0, wE Uk' k = 1,2,3, ....

If d(A) is noncompact, then A is said to satisfy Ditkin's condition at infinity if, whenever x E A, there exists a sequence {xkl C A such that

8.5.

Ditkin's Theorem

A

(b)

xk E

205

Cc(A(A)), k

= 1,2,3, •.•. A

We remind the reader that Z(x) = (T I T E a(A), X(T) = 0) (Definition 7.1.1). It is apparent that, if A satisfies Ditkin's condition at infinity, then A is Tauberian. An immediate corollary of Ditkin's Theorem will show that, if A is asemisimple regular commutative Banach algebra that satisfies Ditkin's condition at each point of A(A) and at infinity when A(A) is noncompact, then a closed set E C A(A) will be a set of spectral synthesis for A provided that bdy(E) contains no nonempty perfect set. Similarly it follows that, if I is a closed ideal in such a Banach algebra, then I is the intersection of the maximal regular ideals containing I, that is, I = k[h(I)], provided bdy[h(I)] contains no nonempty perfect set. For the sake of completeness we recall that, if subset of a topological space X, then bdy(E) = E n E n [X - int(E)], and a set E is perfect if E is every point of E is a limit point of E, that is, lated points.

E is a closed cl(X - E) = closed and E has no iso-

One further lemma is necessary before we can prove Ditkin's Theorem. Lemma 8.5.1. Let A be a semisimple regular commutative Banach algebra, let I be a closed ideal in A, and suppose x E A. Then x belongs locally to I at each T in a(A) that satisfies either of the following conditions: (i) (ii)

T

E int[Z(x)].

T E

Proof.

a(A) - h(I).

Suppose

T E

int[Z(x)].

Then there exists an open

8. Ideal Theory

206

neighborhood U of T such that U c Z(x), and so •x(w) w E U. Thus x belongs locally to I at T.

= •O(w) = O.

Now suppose T ( A(A) - h(I). Then, since A(A) is locally compact, there exists an open neighborhood U of T such that cl(U) is compact and cl(U) C A(A) - h(I). Since cl(U) is compact, h(l) is closed, and cl(U) n h(l) we see from Theorem 7.3.2 that there exists some y E I such that •yew) = 1, w E cl(U). Hence • •• • xy E I and (xy) (w) = x(w)y(w) = x(w), w E U. Therefore x belongs locally to I at T.

="

o

Theorem 8.5.1 (Ditkints Theorem). Let A be a semisimple regular commutative Banach algebra that satisfies Ditkin's condition at each point of A(A) and at infinity when A(A) is noncompact, let I be a closed ideal in A, and suppose x E A is such that h(l) C Z(x). Then (i) If E is the set of T E A(A) such that x does not belong locally to I at T, then E is a perfect subset of h(l) n bdy[Z(x)] = bdy[h(I)] n bdy[Z(x)]. (ii) If bdy[h(I)] n bdy[Z{x)] set, then x ( I.

contains no nonempty perfect

Proof. From Lemma 8.5.1 and the fact that Z(x) is closed we see at once that E C h(l) n (~(A) - int[Z(x)]). However, since h(l) C Z(x), we have bdy[h{I)] nbdy[Z(x)] = (h{I)

ncl[~(A)-h(I)])

n (Z{x)

ncl[~(A)-Z{x)])

=h(l) n (Z{x) ncl[~{A)-Z(x)]) = h{l) nbdy[Z(x)] • h (I)

n [Z (x) n (6 (A) -

int [Z (x)])]

= h{l) n (6(A) - int [Z{x)]).

8.5.

Ditkin's Theorem

207

Consequently E C h(l) n bdy[Z(x») = bdy[h(I)] n bdy[Z(x»). Moreover, from the definition of local membership in an ideal we see at once that E is closed. Thus to show that E is perfect we need only prove that E has no isolated points. So suppose that T € E is isolated. Then there exists some open neighborhood U of T such that cl(U) is compact and (cl(U) - (T) n E = ,. Since E C bdy[Z(x)] c Z(x), we see that T E Z(x). Thus, by Ditkin's condition, there exist a sequence (x k ) • in A and open neighborhoods Uk of T such that xk(w) = 0, wE Uk' k = 1,2,3, •.. , and limkllxxk - xII = O. Let W be an open neighborhood of T such that cl(W) C U. This is possible because A(A) is locally compact. Then cl(W) is compact, A(A) - U is closed, and cl(W) n [A(A) - U] = " whence, since A is normal, • we deduce the existence of some y £ A such that yew) = I, w E cl(W), • and yew) = 0, w ~ A(A) - U. We claim that the elements yxx k , k = 1,2,3, ••• , belong locally to I at each point of A(A) and at infinity.

•

Indeed, since xk(w) = 0, w E Uk' and T E Uk' we see that yxx k belongs locally to I at T; and if w £ cl(U) - (T), then by the definition of E we see that x belongs locally to I at w, whence yxxk belongs locally to I at w, k = 1,2,3, .... Thus • each yxx k belongs locally to I on cl(U). Finally, since y vanishes identically on A(A) - U and cl(U) is compact, we see at once that yxx k ' k = 1,2,3, •.. , belongs locally to I at each w E A(A) - cl(U) C A (A) - U and at infinity. Hence yxxk belongs locally to I at each point of A(A) and at infinity, k = 1,2,3, .•.• Therefore (yxx k ) C I by Theorem 8.4.1. Consequently yx £ I, as I is closed and limkllyxxk - yxil = O. • ••• • But if w £ W, then (yx) (w) = y(w)x(w) = x(w), as y(~) = I, w E cleW). Since T E W, this says that x belongs locally to I at T, contradicting the fact that TEE. Therefore

E

is perfect and part (i) of the theorem is proved.

8. Ideal Theory

208

To prove part Cii) we suppose that bdy[hCI)] n bdy[Z(x)] contains no nonempty perfect sets. Then from part (i) we see that x belongs locally to I at each point of a(A) , as E =~. If aCA) is compact, then x E I by Theorem 8.4.1. On the other hand, if a(A) is noncompact, then, since A satisfies Ditkin's condition • at infinity, there exists a sequence (xk ) C A such that (xk) is contained in Cc (6(A» and limklixxk - xII = O. For each k let Ek be the set of T E a(A) such that xXk does not belong locally to I at T. We claim that Ek = ~, k = 1,2,3, ..•. Indeed, since x belongs locally to I at each point of a(A), we see that, if T (A(A), then there exist an open neighborhood U • • of T and some y E I such that x(w) = yew), w (U. Clearly • • then we must also have (xx k) (w) = (YX k ) (w), w E U, and YXk E I, k = 1,2,3, ... ; that is, xX k belongs locally to I at T, k = 1,2,3, •••• Hence Ek C E = ~, k = 1,2,3, •.••

-

Thus xX k belongs locally to I at each point of 6(A), k = 1,2,3, •... Moreover, since (x k) C CcCaCA», it is apparent that xXk also belongs locally to I at infinity, k = 1,2,3, ...• Consequently by Theorem 8.4.1 we conclude that (XX k ) C I, whence x E I, as I is closed. It should be noted that the hypothesis that A satisfies Ditkin's condition at infinity was only utilized in proving part (ii) of the theorem. Thus Theorem 8.S.1(i) remains valid under the weaker assumption that A satisfies Ditkin's condition only at each point of a(A). Some easy consequences of Ditkin's Theorem are the following corollaries: Corollary 8.5.1. Let A be a semisimple regular commutative Banach algebra that satisfies Ditkin's condition at each point of a(A) and at infinity when a(A) is noncompact. If E C a(A) is a

8.6. LI(G) Satisfies Ditkin's Condition

209

closed set such that bdy(E) contains no nonempty perfect set, then E is a set of spectral synthesis for A. Proof. Let I be any closed ideal such that h(I) = E. If x E k[hel)] = keEl, then obviously E = hel) c Zex), and bdy(E) = bdy[h(I)] ~ bdy[h(I)] n bdy[Z(x)]. Since bdy(E) contains no nonempty perfect set, it follows that bdy[h(I)] n bdy[Z(x)] has the same property, whence we conclude that x E I by Ditkin's Theorem. Hence I = k[h(I)] = k(E), and E is a set of spectral synthesis' D Corollary 8.5.2. Let A be a semisimple regular commutative Banach algebra that satisfies Ditkin's condition at each point of a(A) and at infinity when a (A) is noncompact. If I C A is a closed ideal such that bdy[h(I)] contains no non empty perfect set, then I is the intersection of the maximal regular ideals containing I; that is, I = k[h(I)]. It is perhaps worthwhile mentioning explicitly that, if A is a semisimple regular commutative Banach algebra, then a (A) is compact if and only if A has an identity. This is the content of Corollary 7.3.2. Thus in the theorems of this and the preceding section a(A) is noncompact precisely when A is without identity. In the next section we shall show that LI(G), G being a locally compact Abelian topological group, satisfies Ditkin's condition, and so we may apply Ditkin's Theorem to this algebra. 8.6. L1(G) Satisfies Ditkin's Condition. Before we prove that LI(G), G being a locally compact Abelian topological group, • satisfies Ditkin's condition at each point of a(LI(G)) = G and • is noncompact, we shall establish two lemmas . at infinity when G • and As usual, ij will denote Haar measure on the dual gruop G, • have been so we shall assume that the Haar measures on G and G chosen that Plancherel's Theorem (Theorem 7.2.1) is valid.

210

8. Ideal Theory

Lemma 8.6.1. Let G be a locally compact Abelian topological • be an open neighborhood of the identity y in group, let WC G • 0 G, let 6 > 0, and let E C G be compact. Then there exist a compact set KeG• with nonempty interior and an open symmetric • neighborhood a of Yo in G such that (i) (ii) (iii) (iv) (v)

cl(U)

is compact.

y € int{K). o

~(K

- U) <

K + U - UC

2~(U).

w.

11 - (t,y)1 < 6, tEE and y E K + U -

u.

Proof. As observed at the beginning of the proof of Theorem • to r defined by (t,y) = (t,y), 4.7.5, the mapping from G x G • t E G and y E G, is continuous. Thus for each tEE there • exist open neighborhoods Vt of t in G and Wt of Yo in G such that 11 - (s,y)1 < 6, s E Vt and y E Wt . Since E is compact and (Vt I t E EJ forms an open covering of E, we deduce the

0:

C

existence t l ,t 2 , ... ,tn in E such that E ~=IVtk. Set WI =.W n (Ok=lWtk ). Evidently WI is an open neighborhood of Yo in G, and 11 - (t,y)1 < 6, tEE and y E WI. Moreover, using the local compactness of G and the argument of Corollary 7.2.2, • and a symwe easily deduce the existence of a compact set Kl C G • such that metric open neighborhood U of y in C o

(b)

cleU)

is compact.

If Tt(K l - U) <

2~(U),

then we set

K = Kl .

If this is not

• is discrete, then we set K = (YoJ. Finally, if the case and G • is not discrete, then ~«(y J) = 0, and we readily deduce from G o

8.6. Ll(G) Satisfies Ditkin's Condition

211

the regularity of ~ that there exists some compact set KeG• such that K C Kl n U, Yo E int(K), and ~(K - U) ~ 2~(U). as o < ~(U) < m by Proposition 4.7.3. Furthermore, it is apparent that with these choices of K we have K + U - U C WI C Wand II - (t,y)l < 6, tEE and y E K + U - U.

o

Lemma 8.6.2. Let G be a locally compact Abelian topological • let W be an open neighborhood of w in G, • group, let w E G, and let 1 > O. If f E LI(G) is such that •few) = 0, then there exists some k (LI(G) such that (i) (ii) (iii)

k• is identically one on some open neighborhood of w.

•key)

• - W. = 0, y E G

IIf * kll} < s.

Proof. In view of the fact that [(.,w)g] • (y) = •g(y - w), y (G• and g E LI(G), we may assume without loss of generality that w=Yo. Let 0<6<,/[4I2(1+lI f ll})]. Since fEL 1 (G), there exists a compact set E C G such that

I G-

E If(t)1 dAft) < 6.

..

Moreover, by Lemma 8.6.1 there exist a compact set KeG • such that open symmetric neighborhood U of y in G o

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

(v)

cl{U)

is compact.

Yo (int(K). ~(K

- U) <

K+ U - Uc

2~(U).

w.

11 - (t,y)1 < 6,

t

£ E and

y E K + U - u.

and an

212

8. Ideal Theory

By Plancherel's Theorem (Theorem 7.2.1) there exist k1 ,k 2 £ L2 (G) • • such that kl = XK _ U and k2 = XuI'Il (U) , where, as usual, XK- U and Xu denote pectively. The transform. Set form of k is Theorem 7.2.3. • that key) = 1, Thus it remains

the characteristic functions of K - U and U, restransform involved is, of course, the Plancherel k = kl k2 • Then k € LI(G) and the Fourier trans•k = X _ * [Xu/~(U)], as was seen in the proof of K U The argument of that theorem also shows immediately • - W. Y € K, and • key) = 0,• y E G - (K + U - U) ::> G only to show that IIf * kill < s.

To this end we note first that

= IG

f * k(t)

=

because •f(yo)

= O.

IG

f(s)k(t - s) dA(s) f(s)[k(t - s) - k(t)] dA(s)

(t ( G)

Hence

IIf * kl/l = IGlfG f(s) [k(t - s) - k(t)] dA{s)1 dArt) ~IG If{s)I[IG lTs{k){t) - k{t)l dArt)] dA(s)

I

= G- E If{s)IIlTs(k) - kill dA{s) +

IE If(s)IIITs{k) - kill dA.{s).

Thus, using Plancherel's Theorem (Theorem 7.2.1) and the fact that ~(K - U) < 2~(U)J we see that

I G-

E If{s) 1IITs{k) - kill dA(s) < 2I1 kIl I I G_ E If(s)

< 2611klk2111

< 2611klli21/ k2112

- .

= 2611klil2l1k2112 =

Xu

26I1xK_uIl211~{u)1I2

I dA.(s)

8.6. LI(G) Satisfies Ditkin's Condition

213

= 26[~(K~(U~)]1/2 < 2/26.

In order to estimate the second integral we observe that Ts(k) - k

= Ts (k 1)Ts (k 2) = kI [Ts (k 2)

- klk2

- k2]

+

[Ts(k 1) - kI ]T s (k 2)

G).

(s €

Furthermore, from Plancherel's Theorem we see that, if s (E,

then

= II-=--(s-,.~Jkl - kIII2

= [Ia =

11 - (s,y)1 2xK_u(Y) d~(y)]1/2

[IK- u II -

(s,y)12 d~(y)]1/2

< 6[~(K _ U)]1/2

because 11 - (s,y)l < 6, sEE and y E K + U - U. we see that for each sEE

IITs (k 2)

- k2112 <

6 1/2. [ll (U) ]

Combining these estimates with the facts that and IIk2112 = 1/[~(U)]1/2, we deduce that IITs(kJ - kill < IIk l [Ts (k 2) - k2]11 1

~ IIkI1l2 I1Ts(k2) - k21i2

Similarly,

IIkl1l2

= [~(K _ U)]1/2

II[T s (k I J - kl ]Ts (k 2)lI l

+ +

IITs(k l ) - klIi2IiTs(k2)1I2

< 26[TI(K - UJ]I/2 -

< 2./'26

whence

~(U)

(s E E) J

214

8. Ideal Theory

Consequently we see at once that

IIf

* kill ~ 2/26

+ 2/26

11 f ll l

= 2/26(1 + IIflll)

<

I,

which completes the proof.

o

We are now in a position to prove the main theorem of this section. Theorem 8.6.1. Let G be a locally compact Abelian topological group. Then Ll(G) satisfies Ditkin's condition at each point of • and at infinity when G • is noncompact. G Proof. The assertion that Ll(G) satisfies Ditkin's condition • is noncompact is precisely the content of at infinity when G • and f E Ll(G) is such that Corollary 8.1.3. So suppose wE G •few) = o. If n is a given positive integer. then again appealing to Corollary 8.1.3 we see that there exist some v E Ll(G) such • • n that vn E Cc.(G) and IIf - f * vn III < 1/2n. Clearly f * vn ~ Ll (G) and (f * v ) (w) = O. Next, applying Lemma 8.6.2 to f * v, we n • n deduce the existence of some k E LI(G) such that k (y) = 1. n n y E V. where V is a suitable open neighborhood of w, and such n n that Ilf * v * k III < 1/2n. Set g = v - v * k. Then g E Ll (G) • n. n • • • n n. n n n and gn(Y) = vn(Y) - vn{y)kn(Y) = vn(y) - vn{Y) = 0, Y E Vn" Moreover,

IIf -

f .. gn III

= IIf - f 1 + 1

< 2n

2n

* (vn - v n * kn Jill

1 n

= -.

8.7. Further Remarks on Ideals

215

Consequently limnllf - f * gnU} = 0, and Ll(G) satisfies Ditkin's condition at wand hence at each point of G.

..

C!

Thus Ditkin's Theorem (Theorem 8.5.1) and Corollaries 8.S.l and 8.5.2 are valid for Ll (G). These give us the following result: Corollary 8.6.1. gical group. Then

Let G be a locally compact Abelian topolo-

• (i) If E eGis a closed set such that bdy(E) contains no nonempty perfect set, then E is a set of spectral synthesis for Ll (G) . (ii) If I C Ll(G) is a closed ideal such that bdy[h(I)) contains no nonempty perfect set, then I is the intersection of the maximal regular ideals containing Ij that is, I = k[h(I)). As some concrete applications of this corollary we observe that, • if E eGis closed and bdy(E) is discrete, then E is a set of spectral synthesis for Ll(G). Thus, for example, if G = Rand E = [-1,1], we see that E is a set of spectral synthesis for LIOR) , as bdy(E) = (-1,1). This partially proves the remark made concerning L.Schwartz's example discussed after Malliavin's Theorem (Theorem 8.3.4). The description of the sets of spectral synthesis for Ll(G), G being a compact Abelian topological group, given by Theorem 8.3.3 and Corollary 8.3.2,is also an immediate consequence of the preceding remark. Indeed, in this case we have by Corollary 4.7.5 that • is discrete, and so bdy(E) is empty for every set E C G. • G 8.7. Some Further Remarks on Ideals. In the preceding sections of this chapter we have developed a modest portion of the ideal theory for commutative Banach algebras and examined some of the abstract results in specific Banach algebras. As should be apparent, the study of ideal structure is a very complicated affair, even for

8. Ideal Theory

216

specific algebras, and we shall not pursue it further here. However, before leaving the subject we would like to indicate some standard questions concerning ideals, two of which were mentioned in the introduction to this chapter, and tabulate the results for certain algebras. The proofs of a portion of these results appear in the preceding pages. Definition 8.7.1. Let A be a commutative Banach algebra. A closed ideal I in A is said to be primary if it is contained in precisely one maximal regular ideal in A. Evidently every maximal regular ideal is primary, but the converse may fail. For example~ consider the £losed ideal I in CnCrO,I]), n ~ 1, defined by I = {f

I

f (Cn([O,I]), f(O) = f'CO)

Then the only maximal ideal containing M = {f while clearly

I

~

I

I

= 0).

is

f (Cn(rO,I]), fCO)

= 0),

M.

Obviously a primary ideal is always proper. Some general questions about ideals in a commutative Banach algebra A are the following: 1. Is every proper closed ideal I maximal regular ideal; that is, is hCI) 2. Is every proper closed ideal I the maximal regular ideals containing I; 3.

Is every primary ideal

I

in

A contained in some

~ ~?

in A the intersection of that is, is I = k[hCI)]?

in A a maximal regular ideal?

4. Is every proper closed ideal of the primary ideals containing I?

I

in

A the intersection

8.7. Further Remarks on Ideals

217

The answers to these questions for some particular Banach algebras are contained in the table below.

* ** Algebra Ll(G) ** G a noncompact ** locally compact ** ** Abelian * Question

topological group

Cn([O, 1])

A(D)

n> I

-

C(X) X a compact Hausdorff topological space

I

Yes

Yes

Yes

Yes

2

No

No

No

Yes

3

Yes

No

No

Yes

4

No

Yes

No

Yes

More generally the answer to question I is affirmative either when A has an identity or A is a Tauberian semisimple regular commutative Banach algebra. However, if A is a radical algebra, then the answer to question 1 is negative. For further material on ideal structure in addition to that cited in this chapter the reader is referred to [N,Ri).

CHAPTER 9 BOUNDARIES 9.0. Introduction. Recall that A(D) is the commutative Banach algebra with identity of all those complex-valued functions that are defined and continuous on the closed unit disk D and analytic on int(D). The Maximum Modulus Theorem of analytic function theory asserts that, if f E A(D) , then IIf llCl) = sup, €. rl f (C) (. where r = (e Ie€. D, lei = 1), and there exists some int(D)

,€

such that IIfllCD = If(C)1 only if f is a constant function. A second central result of analytic function theory, the Cauchy Integral Formula, tells us that, if f € A(D), then f(C)

= ~S 2nl r

fez) dz

z -

C

(C E int (D) ) .

In this chapter we shall examine the question of obtaining generalizations or analogs of these classical results in the context of commutative Banach algebras. Actually, it is advantageous to shift the investigation away from commutative Banach algebras or, more precisely, away from the Gel'fand representations of such algebras to a more general class of algebras, which we shall call separating function algebras. In the following sections we shall develop a portion of what can be said about the Maximum Modulus Theorem and the Cauchy Integral Formula for such algebras. The development falls roughly into two parts. In Sections 9.1 through 9.4 we shall concern ourselves primarily with analogs of the Maximum Modulus Theorem. This leads to the introduction of the concept of a boundary for a separating function algebra, which will be seen to be a suitable replacement for r in the study of A(D). We shall investigate three specific boundaries: the ~ilov, 218

9.1. Boundaries

219

Bishop, and Choquet boundaries. In Sections 9.6 through 9.8 our primary, but not our only, concern will be counterparts of the Cauchy Integral Formula. This development leads to the notion of representing measure. From the classical situation, as exemplified in A(D), it should come as no surprise that the notions of boundaries and representing measures are intimately connected. This connection will be perhaps most apparent in Sections 9.6 and 9.7. Between the two portions of the chapter just indicated is a section devoted to several applications of the material on boundaries developed up to that point. However, the range of applications of the material introduced in the chapter is substantial, particularly in the study of function algebras, and we have made no attempt to provide a general introduction to these applications. The reader who is interested in such applications should consult the appropriate references cited in this chapter. 9.1. Boundaries. Consider the commutative Banach algebra with identity A(O) consisting of all those functions defined on the closed unit disk 0 in C that are continuous on 0 and analytic on int(O). Then the classical theory of analytic functions contains a number of results concerning the maximum modulus of elements of A(O). To be precise, the following results are valid. Recall that

r = (C ICE D, Ici = 1). If f E A(D), then there exists some C E r such that IfCC) I = II f Il CD • In particular, we see that IIfU. = suPC E rl fCC) I and that If I assumes its maximum on f. 1.

2. If E c r is closed and if for each f E A(D) there exists some C E E such that If(C)1 = IIfll, then E = r. In particular, CD r is the smallest closed subset of D on which If I assumes its maximum for each f E A(D). 3. If f E A(D) and there exists some If(C)1 = UfU, then f is a constant. CD

C E int(D)

such that

9. Boundaries

220

4. If f (. A(D) is nonconstant and e E int(D) , then for each p > 0 such that (z I Iz - el < p] C int(D) there exists some w, Iw - el < p, such that If(w)1 > If(e)l· In other ,~ords, if f ( A(D) is nonconstant, then I f I cannot have a local maximum at any point in int(D).

The first result or some variation of it is usually referred to as the Maximum Modulus Theorem or the Maximum Principle (see, for example, [A, pp. 133-135; Ru 2 , p. 213]). The fourth result is called the Local Maximum Modulus Theorem. Our main concern in this and the following four sections will be to examine some extensions of these results for A(D) to general commutative Banach algebras. In particular we shall see that there always exist valid analogs of result~ 1 and 2, but result 3 may very well fail to hold. A valid analog of result 4 also exists, but a proof would involve considel'nble machinery which we have not developed. The curious reader is referred to [Ga, pp. 91-93; S, pp. 89-98; Wm 2 ' pp. 52-55] for these generalizations of the Local Maximum Modulus Theorem. There is some advantage in the sequel in working with algebras of continuous functions rather than an arbitrary commutative Banach algebra. Moreover, the Gel'fand Representation Theorem (Theorem 3.3.1) asserts that every commutative Banach algebra A is continuA ously homomorphic to a subalgebra A of C (A(A) that separates o the points of A(A). This theorem will then allow us to translate results about such subalgebras of C (X), X being a locally compact o Hausdorff topological space, into results about commuative Banach algebras. With this in mind we make the following definition: Definition 9.1.1. Let X be a locally compact Hausdorff topological space and suppose A is a subalgebra of C (X). Then A o is said to be a separating function algebra on X if (i) (ii)

A separates the points of X. ZeAl = {t

I

t

E X, f(t) = 0,

f

E A] = ,.

9.1. Boundaries

221

Clearly, if X is compact and Ac C(X) is a subalgebra that separates the points of X and contains the constant functions, then A is a separating function algebra. Furthermore, if A is • a commutative Banach algebra, then A = Ace (a(A)) is a separating o function algebra. A useful observation about such algebras is contained in the next proposition. The proof is left to the reader. Proposition 9.1.1. Let X be a locally compact Hausdorff topological space. If A is a separating function algebra on X, then the topology on X is the weakest topology on X such that all the functions in A are continuous. We conclude this section with several additional definitions. Definition 9.1.2. Let X be a locally compact Hausdorff topological space and let A be a separating function algebra on X. A set E c X is said to be a boundary for A if for each f E A there exists some

tEE

such that

If(t)1

= IIfllCD = sUPsExlf(s)l.

Thus a boundary for a separating function algebra A is a subset of X on which If I assumes its maximum for each f E A. For obvious reasons boundaries are often called maximum modulus sets. If A = A(D), then the Maximum Modulus Theorem for analytic functions says precisely that r is a boundary for A(D) and that r is even the smallest closed subset of D that is a boundary for A(D). We shall see shortly that every separating function algebra has a smallest closed boundary. On the other hand, a smallest boundary need not exist. Definition 9.1.3. Let X be a locally compact Hausdorff topological space and let A be a separating function algebra on X. A point t E X is said to be a peak point for A if there exists some f E A such that If(t)1 = IIfll CD = 1 and If(s)1 < 1, s ~ t. The set of all peak points for A is called the Bishop boundary for A and is denoted by pA.

9. Boundaries

222

It is apparent that, if t € X is a peak point for A, then t belongs to every boundary of A, and so pA is the minimal boundary for A. However, there exist separating function algebras A that have no peak points -- that is, such that pA =~. We shall see an example in Section 9.3. Peak points are also called strong boundary points and unique maximum points (see, for instance, [HOI' p. 61; Ri, p. 141]). The term "strong boundary point", however, is usually applied to points that we shall call weak peak points. The reader is warned that there seems to be no fixed terminology for these concepts and is advised to check the definitions in the individual references cited. 9.2. The §ilov Boundary. The main theorem of this section is that every separating function algebra possesses a unique minimal closed boundary, that is, a closed boundary that is contained in every closed boundary. We shall also show that this is true, in a suitable sens~of any commutative Banach algebra and that, if such an algebra A is either regular or self-adjoint, then the minimal closed boundary is precisely A(A). This unique minimal closed boundary will be called the §ilov boundary. I

Theorem 9.2.1 (§ilov). Let X be a locally compact Hausdorff topological space. If A is a separating function algebra on X, then A has a unique nonempty minimal closed boundary. Proof. Let E denote the family of all closed boundaries for A. Clearly E ~~, as X € E. We introduce a partial ordering in E by setting EI > E2 , EI ,E 2 E E, if and only if El C E2 . If (E) is a linearly ordered subset of E, then obviously E = n E a 0 a a is a closed subset of X. We claim that E is also a boundary for

A.

o

Indeed, suppose f E A and f is not identically zero. Define Ef C X as Ef = {t l t E X, If(t) 1 = IIflt'). We claim that Ef is a nonempty compact subset of X. Since A c C (X), there exists o

9.2. The §ilov Boundary

223

some compact set K c: X such that If(t) 1 < IIfll CD/2. t E X - K. Thus f restricted to K is continuous, and so Ifl assumes its maximum modulus on K. Hence Ef is a nonempty closed subset of K, and so Ef is a nonempty compact subset of X. Moreover, since each Ea is a closed boundary for A, we see that Ef n Ea is a nonempty compact subset of K for each a. Since the family {E) is lineartl ly ordered, we see that the family {E f n EaJ has the finite-intersection property, whence we conclude that na(Ef n Ea1 = Ef n Eo is nonempty. In particular, this also shows that E itself is nono empty and that, if t E Ef n Eo c: E, then If(t)l = IIfll. Evident0 CD ly, if f E A is identically zero. then there exists some tEE o such that If(t)l =Ufll. Hence we see that E0 = naa E is a closed CD boundary for A. Thus every linearly ordered subset of E has an upper bound, and so, appealing to Zorn's Lemma [D5 l • p. 6], we deduce the existence of a maximal element in E. call it E. Clearly E is a minimal closed boundary for A, and to show that it is unique it suffices to prove that, if F is any closed boundary for A, then E c: F. To see t~is suppose there exists tEE - F. Now, by Proposition 19.1.1, the topology on X is the weak topology genera~ed by the elements of Aj that is, a neighborhood base for the topology of X at the point t consists of sets of the form

where s > 0 and f l ,f 2, •.. ,fn in A are arbitrary. In particular, since F is closed, there exist some s, 0 < s < 1, and f l ,f2, •.• ,fn in A such that t E U(tjeifl'f2 ••.• ,fn) = U and U n F = ,. Moreover, we may assume, without loss of generality. that IfkCs) - fk(t)1 < 1, k = 1,2, •.•• n; sEX, on the following grounds: If sUPk=1,2, •.• ,n[suPsEXlfk(s) - fk(t)il = M> 1, then set gk = fk/M, k = 1,2, .••• n. Clearly gk £ A, Igk(S) - gk(t)1 < 1, k = 1,2, ••. ,n; 5 ~ X. and U(t;c/M;gl,g2, .•• ,gn) c: U(t;c;f l ,f2 ••.• ,fn).

9. Boundaries

224

Moreover, since E is a minimal closed boundary, there evidently exists some f E A such that IIfli CD = 1 and If(s)1 < 1, sEE - U. Note that, if E - U is empty, then the second assertion is trivially satisfied. Let g E A be a sufficiently high power of f such that Ig(s) I < I, sEE - U. Clearly one still has Then we see that, for each k = 1,2, •.. ,n,

= 1.

IIgli CD

(s E U)

<e

and

<

(s E E - U),

I

whence Ig(s)fk(S) - g(s)fk(t)1 < I, sEE. But E is a boundary for A, and so we have 19(s)fk(s) - g(s)fk(t)1 < I, k = 1,2, ... ,n and sEX. However, since F is a boundary for A, there exists some w E F such that 19(w) 1 = IIgli CD = 1. Consequently

{

<

that is,

w C U,

contradicting the fact that

Therefore E C F, dary for A.

and

(k

I

Un

F

= l,2, •.• ,n), = ,.

E is the unique minimal closed boun-

o

A proof of this theorem that does not depend on Zorn's Lemma is also available (see, for examvle, [5, pp. 37-39]). This theorem provides another illustration of the advantages in requiring the scalar field to be C, since if one considers separating function algebras over the real numbers, then a unique minimal closed boundary need not exist. For further details see [Ri; pp. 134 and 135, 311 and 312].

9.2. The ~ilov Boundary

225

In view of the existence of a unique minimal closed boundary for a separating function algebra we make the following definition: Definition 9.2.1. logical space and let

Let

X be a locally compact Hausdorff topo-

A be a separating function algebra on

The unique minimal closed boundary for dary and will be denoted by cA. It is apparent that

pAc

X.

A is called the ~ilov boun-

cA.

Two corollaries of the proof of Theorem 9.2.1 are worth mentioning.

The details are left to the reader.

Corollary 9.2.1. Let X be a locally compact Hausdorff topological space. If A is a separating function algebra on X, then cA is the intersection of all the closed boundaries for A. Corollary 9.2.2.

Let

logical space and let If t E X, (i)

(iii) i

f E A such that

IIfll

CD

=I

i,

then there exists some

t

in

X,

then there

If(s)1 < 1, sEX - U.

and

If U is an open neighborhood of

< 1,

If(s)1 <

X.

E cA.

If U is an open neighborhood of

exists some

o<

A be a separating function algebra on

then the following are equivalent: t

(ii)

X be a locally compact Hausdorff topo-

t

in

f E A such that

X and IIfll

CI)

=I

and

sEX - U.

The next result provides us with a sufficient condition that

cA = X. Theorem 9.2.2. gical space and let A is norm dense in Proof.

Let

Let X be a locally compact Hausdorff topoloA be a separating function algebra on X. If Co (X), then oA = x.

t E X and suppose U is an open neighborhood of

t.

9. Boundaries

226

Since C (X) is a regular commutative Banach algebra, there exists o some g E Co (X) such that get) = land g(s) = 0, sEX - U. Let f E A be such that IIf - gil CD < 1/2. Then it is easily verified that If(s), < 1/2, sEX - U, by Corollary 9.2.2, that

and ,f(t), > 1/2, from which it follows, t E oA. Hence oA = X.

o

Corollary 9.2.3. Let X be a locally compact Hausdorff topological space and let A be a separating function algebra on X. If A is closed under complex conjugation, then oA = X. If A is a commutative Banach algebra, then, as already noted, the Gel'fand representation A of A is a separating function algebra on ~(A). This observation prompts the following definition:

..

Definition 9.2.2. Let A be subset E of ~(A) is said to be boundary for A, and a point T ~ for A if T is a peak point for

..

a commutative Banach algebra. A a boundary for A if E is a A(A) is said to be a peak point A.

..

Theorem 9.2.1 then shows at once that each commutative Banach algebra A possesses a unique minimal closed boundary, the §ilov \ boundary for A. This boundary will also be referred to as the §ilov boundary for A, and we shall denote it by oA, that is, oA = oA. For commutative Banach algebras where the Gel'fand transformation is the identity mapping, for instance, C (X) or A(D), o .. this is a natural notation since C (X) = c (X) and A(D) = A(D). o 0 Similarly the set of all the peak points for a commutative Banach algebra A will be called the Bishop boundary for A, and we shall denote it by pA = pA.

..

..

..

..

Corollaries 9.2.1,9.2.2, and 9.2.3 and Theorem 9.2.2, of course, have valid analogs for any commutative Banach algebra. For example, the first and second portions of the next theorem follow immediately from the first part of the proof of Theorem 9.2.2 and from Corollary 9.2.3, respectively.

9.3. Examples of Boundaries

227

Theorem 9.2.3. Let A be a commutative Banach algebra. Then aA = A(A) if either of the following two requirements is fulfilled: (i) (ii)

A is regular. A is self-adjoint.

•

Recall that A being self-adjoint means precisely that A is closed under complex conjugation (Definition 5.1.1). 9.3. Some Examples of Boundaries. In this section we wish to describe aA and pA for some specific separating function algebras.

Example 9.3.1. Let X be a locally compact Hausdorff topological space. Then C (X) is a semisimple regular commutative o Banach algebra, and so from Theorem 9.2.3 we see that aCo(X) = A(Co(X» = X. If X is oompact, then we also have pCo(X) = pC(X) = X, provided that X is metrizable. Indeed, if X is compact and metrizable, then for each t E X there exists a sequence of open neighborhoods Uk of t such that Uk+ l C Uk and n;=lUk = {t). Since X is a compact Hausdorff topological space, it is a normal . topological space [W 2 ' p. 83], and so there exist open neighborhoods Wk of t in X such that cl(Wk) C Uk' k = 1,2,3,... [W2 , p. 49]. Then, by Urysohn's Lemma [W 2 , p. 55], there exist functions fk ( C(X), k = 1,2,3, ••• , such that (i)

fk(s) = 1, s E cl(Wk).

(ii)

fk(s) = 0, s ( X - Uk.

(iii)

o ~ fk(s)

~

1, sEX.

Evidently f = tk=lfk/2 k belongs to C(X) and f(t) = 1. Moreover, if sEX and s ~ t, then there exists some positive integer no such that s ~ Uk' k > no' Consequently fk(s) = 0, k>n, ~ o

9. Boundaries

228 f(s) Therefore t

=

no

~

f k (s)/2

k

<

no

~

1/2

k=l k=l is a peak point for C(X),

k

=1 and so

pC(X) = X.

We also see at once from Theorem 9.2.3 that 3A = ~(A) when A is either Cn ([a,b]), n ~ 0; L1 (G), G being a locally compact Abelian topological group; or AC(f), since all these algebras are regular. Example 9.3.2. The classical theory of analytic functions, as indicated in Section 9.1, shows at once that oA(O) = f. Moreover, in this case we once again have pA(O) = f on the following grounds: If C ~ r, then the function fez) = (C + z)/2, z (0, obviously belongs to A(O), and If I is easily seen to have a unique maximum at z = C. Thus each point of r is a peak point for A(O), and since pA(O) C oA(O), we conclude that pA(O) = r. Example 9.3.3. Next consider the set O2 C e2, defined by 02 = ((z,w) 1 Z,w ~ c, Izl ~ 1, lwl ~ 1), and let P(02) denote the uniform closure on 02 of the algebra of all polynomials in two complex variables -- that is, polynomials of the form n "k ~ a"k zJw (aJ"k ~ C; n = 0,1,2, .•• ). j , k=O J Then it is readily seen that P(02) is a commutative Banach algebra with identity where the algebra operations are the usual pointwise ones and the norm is the supremum norm over O2 . Moreover, it is evident that P(02) is a finitely generated Banach algebra generated by the polynomials Pl(z,w) = z and P2(z,w) = w, (z,w) E 02' Thus from Theorem 4.5.1 we see that ~(P(02» can be identified with 02' and the Gel'fand transformation on P(02) is the identity mapping. We claim that oP(02) = PP(02) = ((z,w) 1 (z,w) E e2 , Izl = lwl = I).

Indeed, suppose f E P(02) and suppose (qk J is a sequence of polynomials such that limkllf - qkilCl) = o. Then we see that, for each w E 0,

9.3. Examples of Boundaries

229

f (z) = f(z,w) = lim qk(z,w)

(z ( 0)

k

w

and that the convergence is uniform. Since the polynomials qkC·,w) are analytic for each wED, we conclude that f E A(D). Consew quently from Example 9.3.2 we deduce that IfwI attains its maximum on (z I z (C, Izi = 1] for each w E o. Similarly, if f (w) = z f(z,w), w E 0, then f E A(O) and If I attains its maximum on z z (w I w E C, Iwl = 1]. Combining these two observations, we conclude that, if f E P(02)' then If I attains its maximum on ({z,w) I (z,w) E ~2, Izl that is,

= Iwl = I};

ap(D 2) c (z,w) I (z,w) E ~2, Izi

= lwl

= I).

However, if (e,w) E ~2 and lei = Iwl = 1, then it is easily seen that f(z,w) = (C + z)(w + w)/4, (z,w) E D2, is an element of P(D 2) and that If I has a unique maximum at the point (C,w). Thus every such point in D2 is a peak point for P(D 2), and so ap(D 2) = PP(02) = ((z,w) I (z,w) (C2 , Izi = Iwl = IJ. There are two observations worth making about this example. First, we see that apeD?) is a proper subset of the topological boundary of ~(P(D2)) = D2 since ~

bdy(D 2)

= {{z,w)

I (z,w) E e2 , Izi

=1

or

lwl

= I}.

Second, there exist nonconstant functions f in P(D 2) such that If I assumes its maximum off the ~ilov boundary. For instance, the generators for P{D 2 ) are examples of two such functions. Thus the classic analytic function theorem which asserts that, if f E A(D) and If I attains its maximum on int(D) , then f is a constant has no valid analog for arbitrary commutative Banach algebTas or separating function algebras. Example 9.3.4.

For our next example we set

x

=

((z,t)

I

zED,

t

E [0,1])

9. Boundaries

230

with the topology induced on X as a subset of ~3 and let A be the subalgebra of C(X) consisting of all the f (C(X) such that f(·,O) E A(O) and f(O,.) E C([O,I]). Clearly X is a compact Hausdorff topological space and A is a separating function algebra on X that contains the constants. We claim that 3A whereas

pA

= ((z,O)

= 3A -

= 1)

I Izl

U ((O,t)

I

t ( [0,1]),

((0,0)).

Indeed, suppose for each g

E A(O) we define f g (A by

fg(z,t) = g(z)

for

(z,t) E X; t = 0,

= g(O)

for

(z,t) E X; z

f g (z,t)

and for each h E C([O,I])

we define

= 0;

0 < t < 1,

fh E A by

fh(z,t)

= h(O)

for

(z,t) E X; t

= 0,

fh(z,t)

= h(t)

for

(z,t) ( X; z

= 0;

°< t < 1.

Using the facts that oA(O) = pA(D) = f, oC([O,I]) and that oA is closed, we see easily that oA

= ((z,O)

I Izl

= 1)

= pC([O,I]) = [0,1]

U ((O,t) I t E [0,1])

and that every point in oA - (CO,O)) is a peak point for A. However, (0,0) is not a peak point for A. If it were, then there would exist some f E A such that If(O,O)1 = IIfll CD = 1 and If(z,t)1 < 1, (z,t) EX, (z,t) ; (0,0). But then g(z) = f(z,O), z E 0, would be an element of A(O) for which Ig(O)1 = 1 and Ig(z)! < 1, z E 0, Z ; 0, contradicting the fact that pA(O) = f. Consequently,

pA

= oA -

((0,0)).

Example 9.3.5. Finally we wish to give an example of a separating function algebra that has no peak points -- equivalently, such that the Bishop boundary is empty. Let A be an uncountable set

9.3. Examples of Boundaries

231

and for each a € A let I = [0,1]. Then the topological product a space X = fiaE:AIa is a compact Hausdorff topological space and A = C(X) is a separating function algebra on X. As seen in the first example of this section, we have oC(X) = X. But we claim that pC(X) =~. To see this it suffices to show that there exist two disjoint subsets EO and El of X that are boundaries for C(X). We shall show that the sets EO and El in X that consist, respectively, of all those points t = (t ) E X such that t = 0 a a or t = 1 for all but countably many a E A have the indicated a property. Evidently EO n El =~, so we need only show that EO and El are boundaries for C(X). To this end, for each a E: A we denote by p the projection a of X onto I; that is, p (t) = t , t E X. Clearly each p E C(X), a a a a and we let W denote the subalgebra of C(X) generated by the projections p, a E A, and the constants. That is, W is the subala gebra of C(X) consisting of all polynomials in the projections p , a E A. It is easily verified that W is a subalgebra of C(X) a that separates the points of X, contains the constant functions, and is closed under complex conjugation. Hence, by the Stone-Weierstrass Theorem [L, p. 333], we conclude that W is norm dense in C(X). Consequently, if f E C(X), then there exists a sequence (qk) C W such that li,\lIf - qkllao = O. However, each qk is a polynomial in the projections p, a E A, and so a moment's refleca tion reveals that qk is completely determined by the Pals for only a finite number of a E A. That is, for each k there exists a finite subset -l( ~ of A such that, if s,t E X and pa (s) = pa (t), a E:~, then qk(s) = qk(t). Let Ao = U;=l~' Then Ao is a countable subset of A and, if s,t E X and p (s) = p (t), a E A , a a 0 then qk(s) = qk(t), k • 1,2,3, .•.. Furthermore, it follows immediately from this that, if s,t E: X and p (s) = p (t), a E A, then a a 0 f(s) = f(t), since li~lIf - qkllao = o. But now suppose t = (t g ) E X is such that If(t)l = IIfli• • Such a point exists since 3C(X) = X. Then define s = (s) and a

9. Boundaries

232

w = {w J in a

X

by s s

ex

= t ex

for

a ( A, 0

a = 0 for a E A - A0 ,

and w a

=ta

w = I

a

for for

a EA , 0

a E A-

A. • 0

p (s) = p (w) = p (t) = t , a a a a a (A, whence If(s)1 = If(w)1 = IIfli. Thus, since f E C(X) is o e arbitrary, we see that EO and EI are both boundaries fOT C(X), and so pC(X) = ~. Obviously s E EO

and w (E I ,

and

9.4. Extreme Points, the ~ilov Boundary, and the Choquet Boundary. If A is a separating function algebra on a locally compact Hausdorff topological space X, then the development of the preceding sections has shown that A always possesses a minimal closed boundary, but it mayor may not have a minimal boundary. The main purpose of this section is to introduce a new boundary for separating function algebras on compact Hausdorff topological spaces that contain the constants. This boundary. which we shall call the Choquet boundary, always exists and lies between the Bishop and the ~ilov boundaries. Moreover, the ~ilov boundary will be seen to be the closure of the Choquet boundary. The existence of the Choquet boundary will be established by the first theorem of this section. Since the proof of this theorem depends on applications of the Krein-Mil 'man Theorem, we wish to recall to the reader some relevant terminology and notation. More detailed discussion can be found in [L, pp. 317-326]. Also, before proving the indicated theorem, we shall establish two lemmas that will be useful in the course of the proof.

9.4. Extreme Points

233

In general, if V is a normed linear space, we denote by V* the Banach space of all the continuous linear functionals on V. The weak* topology on V* will be dentoed by rw*, and V* with this topology by the pair (V*,rw*). If E C V*, then co(E) denotes the convex hull of E; that is, co(E) is all the convex linear combinations of the elements of E. The closure of co(E) in (v*,rw*) will be denoted by co(E), and will be called the closed convex hull of E. If E C V* is convex, then a point x* E V* is said to be an extreme point of E if, whenever xi,xi ( E we have and 0 < a < 1 are such that x* = axi + (1 - a)x x* = xi = xi· The set of extreme points of E will be denoted by ext(E). The space (v*,rw*) is a locally convex topological linear space and the Krein-Mil'man Theorem [L, p. 322] implies that, if E C V* is nonempty, convex, and weak* compact, then ext (E) ; , and E = co[ext(E)]. Our applications of the Krein-Mil 'man Theorem in the theorem below will be to subsets of the spaces V* = C(X)* and V* = A*, where X is a compact Hausdorff topological space and A is a separating function algebra.

2,

The first of the two lemmas is the following result. We recall that, if X is a locally compact Hausdorff topological space, then M(X) is the Banach space of all bounded regular complex-valued Borel measures on X. If ~ E M(X) is positive, we shall write ~ ~

o.

Lemma 9.4.1. Let X be a compact Hausdorff topological space. If ~ (M(X) is such that

then

~ ~

o.

Proof. show that

Since

~

is a regular measure, it clearly suffices to

Ix

f(5) ~(s)

> 0

9. Boundaries

234

for each f E C(X) a function and let

such that

0

f(s) < 1, s £ X.

~

Let

f

be such

Ix f{s) d~(s) = a + ib. where a. b E JR. For each f(s) + ib~. Then

~ E lR

f~ E C (X)

define

If~(s)12 = If{s)12 < 1

+

+

by

f~ (s)

=

b2~2

b2~2

(s £ X).

whence we have IIx f~(s) ~(s),2 < [Ix If~(s)' dl~l(s)]2

~ (1

=1

+ +

b2~2)[Ix d'~'(S)]2

b2~2.

But we also see that IIx f~(s) ~(s),2

= IIx

f{s) d~(s)

+

= la + ib(l +

~)12

= a2 +

~)2.

b2(1

+

ib~Ix ~(S)12

22222 . Thus we have a + b (1 +~) < I + b ~ • ~ E JR. from WhICh we deduce via some elementary calculations that 2b2~ ~ I - (a 2 + b 2). ~ E lR. However, such an inequal i ty can be val id for all ~ E IR only if b = 0, and so we conclude that Ix f(s) ~(s) = a. Furthermore, since

we see that

111 - fll CD -< 1 and

9.4. Extreme Points

235

11 - al = IIx d~(s) - Ix f(s) ~ III - fllCD Ix dl~1 (s)

d~(s)1

< 1, whence we conclude that

a > O.

Therefore

~

> O.

o

Before we give the second lemma we need to introduce another definition. Definition 9.4.1. Let X be a locally compact Hausdorff topological space and suppose ~ E M(X). Then ~ is said to vanish on the open set U c X if

Ix

f(s) d~(s)

=0

for every f E C (X) that has compact support contained in U. The o support of ~ is the complement of the largest open set on which ~ vanishes. The existence of the support of ~ E M{X) can be established by a routine application of Zorn's Lemma. If ~ ~ 0, then it is evident that the support of ~ is a nonempty closed subset of X. Lemma 9.4.2. and let ~ E M(X)

Let X be a compact Hausdorff topological space be such that II~II

If f E C(X)

= Ix

d~(s)

= 1.

has the property that

Ix

f(s)

d~(s) = ~~E}

IE f(s)

d~(s)

for each Borel set E C X such that 0 < ~(E) < 1, then equal to a constant almost everywhere with respect to ~. Proof.

From Lemma 9.4.1 it is apparent that

~

> O.

f

is

Let

t

EX

236

9. Boundaries

be a point of the support of ~ and let e > O. From the continuity of f~ the regularity of ~, and the definition of the support of ~ we See that there exists an open neighborhood U of t such that

o<

~(U) < I

If(t) -

and such that

If(t) - f(s)l <,~ s

Ix res) d~(s)l = ,~~)SU ~ .. ~)

Iu

f(t)

d~(t)

If(t) ..

-

£ U.

Thus

~~u)fu f(s) d~(s)l

f(s) I d.. (s)

< s. Since

£

> 0 is arbitrary, we conclude that f(t)

= JX f(s)

for each t in the support of everywhere with respect to ~.

~.

dlJo (s)

Hence f

is a constant almost

o

Now suppose that X is a compact Hausdorff topological space and that A is a separating fUnction algebra on X that contains the constants. In this case we shall generally denote the function in A that is identically one on X by the s)~bol 1. The context will make clear whether we mean this function or the number "1". If t 'X and we set T(t)(f) = TtC£) • f(t). f 'At then it is apparent that T(t) = 'rt E A'" and that Ih't":: 1't (1) 1. It is easily seen that the mapping ~: X - A* defined above identifies X with a subset TeX) of (x* 1 x* E A*, IIx*1I = x*(l) = 1]. Moreover, we shall see that as this latter set is a nonempty weak* compact convex subset of A*. it has extreme points by the Krein-Millman Theorem, and these ext~eme points all belong to T(X). The set of t E X such that "t is an extreme point of (x" I x* E A*, Ux*1J = x* (1) = 1) is a boundary for A whose closure is aA. This is the main conclusion of the following theorem: 'CO

Theorem 9.4.1 (Bishop-deLeeuw). Let X be a compact Hausdorff topological space and let A be a separating function algebra on X that contains the constants.

9.4. Extreme Points

237

If T : X ~ A* is defined by T{t)(f) = Tt(f) = f(t), t (X and A, then T is a homeomorphism of X onto T(X) when the latter is considered as a subset of (A*,rw*). (i)

(ii)

f'

CO[T(X)] =

{x* 1 x*

E. A*,

IIx*1I

= x*(l)

closure is with respect to the weak* topology (iii)

ext({x·

I

x* E A*, lix*U

= x*(l)

= IJ

rw*

l

A*.

= 1)) C T(X).

(iv) aA is equal to the closure in (A*,rw*) equivalently, aA is the closure in X of {t

on

whe-re the

of ext(co[T(X)]);

t £ X. Tt ' ext(co[T(X)]l].

Proof. From the definition of the weak* topology on A* and the fact that A separates the points of X it fOllows a~ once that T: X ~ A* is injective and continuous. Since the weak* topology is Hausdorff and X is compact, we conclude that T is a homeomorphism of X onto T(X) C (A*,fW*). This proves part (i) of the theorem. To prove part (ii) we first note that IITtU = 'ftCl) = 1, t E X, and so T(X) C (x1t ' x* EA."', IJx*1I = x*(l) = 1] = Moreover, it is easily verified that B~ is convex and weak* closed. Hence CO[T(X)] C B~. However, if x* E A* and IIx*1J = x*(l) = 1, then from a consequence of the Hahn-Banach Theorem [L, p. 87], we see that there exists sOJle y* E CeX)* such that llx*U -= lIy*1I and y* (f) == x* (f) J f E A.. Furtaermore, Ily*1I = IIx*U == x* (1) = y* (1) = 1. Consequently, by the Riesz Representation Theorem for C{X)* [L, p. 109], there exists a unique ~ E M(X) such that

BI.

y*(f)

= Ix

and

Thus, by Lemma 9.4.1,

~

> O.

f(5) ~(s)

(f £ C(X))

9. Boundaries

238

Now we recall that the extreme points of the closed unit ball in M(X), that is, of the set (v I v E M(X), IIvll < 1), are precisely those measures of the form aCt' t (X, where a (~, lal = 1, and 6t E MeX) is the measure with unit mass concentrated at t; that is, if E C X is a Borel set, then 6t (E) = 1 if tEE and 6t (E) = 0 if t ~ E. A proof of this fact is available, for instance, in [L, p. 338]. Since the closed unit ball in M(X) is weak* compact and convex, by the Banach-Alaoglu Theorem [L, p. 254], we deduce, from the Krein-Miltman Theorem [L, p. 322], that it is the weak* closed convex hull of its extreme points. Combining these observations with the fact that ~ > 0 and II~II = 1, one easily verifies that ~ is a weak* limit of convex linear combinations of the extreme points (6 t I t E xl. However, since the restrictions to A of the functionals determined by ~ and 6 t , t E X, are just x* and Tt , t E X, respectively, it follows immediately that x* is a weak* limit of convex linear combinations of the Tt' t E X. That is, x* E -COrTeX)], and so -CO[T(X)] = B11 • Next we shall show that ext(B II ) C T(X). We observe that, since Bi is weak* closed and norm bounded, it must be weak* compact by the Banach-Alaoglu Theorem, whence we conclude from the Krein-Mil'man Theorem that ext(B 1l ) ~~. If x~ E ext (B II ), then, by the argument used in proving part (ii), we know that there exists a measure ~o E M(X), lI~oll = 1, ~o ~ 0, x~(f)

such that

= Ix

f(s) ~o(s)

If E C X is any Borel set such that

0<

(f ~o(E)

E A).

then define

< 1,

xi,xi on A by

(f 1

E A).

It is easily verified that xi and xi belong to B1 . Moreover, it is obvious that x* = ~ (E)x 1* + [1 - ~ (E)]x 2*. Consequently, 1 0 0 0 since x~ E ext(B I ), we have x~ = xi = xi·

9.4. Extreme Points

239

Thus, in particular, we see that (f Eo A)

for each Borel set E in X such that 0 < ~ (E) < 1. By Lemma o 9.4.2 we conclude that each f E A is constant almost everywhere with respect to ~. Since A separates the points of X, it folo lows at once that the support of ~ must be a single point, and o so, since ~0>- 0, there exists some t E X such that ~o = 6t . Thus x~(f) =

whence we have x~

Ix

= Tt .

f(s) d~o(s) = f(t) = Tt(f)

(f E A)'

Hence ext(B~) C T(X).

Finally, suppose F A* - C is a linear functional that is weak* continuous; that is, F is a continuous linear transformation from (A*,rw*) to ~. We claim that

Note that the supremum is finite, as F is continuous on the weak* compact set B~. Clearly the left-hand side of the equation is always greater than or equal to the right-hand side. Suppose that the inequality is strict, let p > 0 be such that sup lIF(x*)1 > x* E 8 1

p

>

sup I IF(x*)I, x* E ext (B l )

and let x* E Bli be such that IF(x*)1 > p. By the Krein-Mil'man o _ 1 0 1 Theorem, we have that co[ext(B I )] = Bl , and so there exist ~ E ext(B~) and a k E~ a k > 0, k = 1,2,3, ... ,n, such that ~=lak = 1

and n IF(x~)' IF(x*) - F( E a x*)1 < 2 o k=l k k

Hence, on the one hand,

p

9. Boundaries

240

> and~

on the other

p~

hand~

n n IF( E akx,) I < ~ akIF(x,ll

k=l

k=I

< p. This contradiction shows that

for each weak* continuous linear functional

F on

A*.

it is apparent that~ if F is a weak* continuous linear functional on A*~ then we also have Furthermore~

=

sup IIF(x*)1 x* €B 1

sup 1 IF(x*)I, x* ( c 1 [ext (B l )]

1 where cl[ext(B~)] denotes the weak* closure of ext(B I ). Moreover, since weak* continuous linear functiona1s separate the points of A* [L, p. 240], it is easily verified that cl[ext(B~)] is the smallest closed subset E C T(X) C Bl such that 1

sup llF(x*)1 x* (B 1

=

sup IF(x*)1 x* (E

for each weak* continuous linear functional

F.

Now, however, each weak* continuous linear functional F on A* is of the form F(x*) = x*(f), x* (A*~ for some f E A [t, p. 238]. Thus, if f E A and F(x*) = x*(f), x* (A*, then

9.4. Extreme Points

241

IIfli

CD

:: sup If (t) I €X

t

= ::

sup ITt(f)1 Tt ~T(X) sup Tt

IF(Tt)1

€ T (X)

::

sup 1 IF(Tt)l Tt E cl[ext(B l )]

::

sup 1 IF(Tt)l Tt E ext (B 1)

=

Tt

sup 1 If(t)l· ~ ext (8 1 )

Consequently we see that, if f E A, IIfllCD :: ::

then

sup -1 1 If(t) 1 t E cl (T [ext (B 1) ] ) _lsuP 1 If(t)l, t ( T [ext (B l )]

and that cl(T -1 [ext(B 11)]) is the smallest closed subset of X for which the identity is valid. Therefore we see that

oA that is,

I = Cl(T -1 [ext(Sl)])::

-cl(T - 1 [ext(co[T(X)])])~

oA is the closure in X of (t I t E X, Tt € ext(co[T(X)])).

This completes the proof of the theorem. The last portion of the proof reveals that

is a boundary for

A that always exists.

make the following definition:

With this in mind we

o

242

9. Boundaries

Definition 9.4.2. Let X be a compact Hausdorff topological space and let A be a separating function algebra on X that contains the constants. If for each t E X we set Tt(f) = f(t). f E A. and B~ = (x* I x* E A*. IIx*1I = x*(l) = 1), then xA = (t I t E X. Tt E ext(B~)} is called the Choquet boundary for The next corollary follows immediately from Theorem 9.4.1. Corollary 9.4.1. Let X be a compact Hausdorff topological space. If A is a separating function algebra on X that contains the constants. then (i)

(ii) (iii)

xA

~

,.

XA is a boundary for

A.

cl(XA) = aA.

From our discussion of boundaries for specific algebras in the preceding section we see at once that XC(X) = X, X being a compact Hausdorff topological space, xA(D) = f. and xP(D 2) = pP(D 2) = ap(D 2). The Choquet boundary can, however, be a rather complicated set. If X is metrizable, then xA is always a G6-set, but if X is not metri:able, then ~ may not even be a Borel set (see, for instance, [5, pp. S4 and 55, 138 and 139]). In Sections 9.6 and 9.7 we shall discuss a number of equivalent descriptions of the Choquet boundary in the case that A is norm closed. In view of Definition 9.2.2 and Theorem 9.4.1. it is evidently meaningful to consider the Choquet boundary XA = XA• whenever A is a commutative Banach algebra with identity. In particular, the Choquet boundary for such an algebra always exists. Before we contiftue an investigation of the Choquet boundary we wish to look at several consequences of the existence of the §ilov and Choquet boundaries.

A.

9.S. Applications of Boundaries

243

9.S. Some Applications of Boundaries. In this section we wish to prove some results about separating function algebras and commutative Banach algebras whose proofs utilize the notion of a boundary. Theorem 9.S.l. Let X be a space and let A be a separating tains the constants. If T : A ~ A such that T(l) = 1, then T

compact Hausdorff topological function algebra on X that conA is a linear isometry of A onto is an algebra isometry of A onto

A. Proof. Let T*: A* ~ A* denote the adjoint of T, that is, T*(x*)(f) = x*[T(f)], x* E A* and f E A. It is easily seen that T* : A.* -- A.* is also a surjective linear isometry. Moreover, since T(l) = 1 and T*[x*(l)] = x*[T(l)] = x*(l), x* € A*, we see that T* maps B~ = (x* I x* E A*, \lx*1I = x*(l) = lJ onto itself. From this observation it is readily deduced that T* maps ext (B 1l ) onto ext(B 11). Now, if t E XA, then Tt E ext(B~), and so there exists some s E XA such that T*(T) = Ts since ext(B 1l ) C T(X). Consequently t for each f,g E A we have T(fg)(t)

= Tt[T(fg)]

= Ts (fg) = (fg) (5) = f(s)g(s)

= Ts(f)TS(g) = T*(Tt ) (f)T*(Tt ) (g) =

T(f) (t)T(g) (t).

9. Boundaries

244

Since

t E XA

is arbitrary, we conclude that for each

f,g E A we

have T(fg)(t) = T(f)(t)T(g)(t), t E xA. whence T(fg)(t) = T(f)(t)T{g){t), t E X, as XA is a boundary for A. Hence T is an algebra isometry.

o

Next we wish to prove a result about topological zero divisors in separating function algebras and in commutative Banach algebras. Clearly a separating function algebra is a normed algebra, and so the notion of topological zero divisors is meaningful in this context. The results on topological zero divisors will be corollaries of the following theorem: Theorem 9.5.2.

Let

X be a locally compact Hausdorff topolo-

A be a separating function algebra on

gical space and let f E A, then

inf If (s) I s E: oA

X.

If

IIfgll m = inf IIglim gEA g~ 0

Proof.

If

g

E A and

t E oA be such that

19(t)1

g

is not identically zero, we let

= IIglimo

Then

inf If(s)1 < If(t)l -

s E: oA

Since this holds for each

g E A, g

inf If(s)l < a s E: oA

~ 0,

=

we conclude that Ii f

inf

gli m

I' II

g E A Ig

.

Q)

g~O

If

a

= 0,

the proof is complete, so we may assume that

a > O.

Let

E = (s I sEX, If(s)1 > aJ. Evidently E is a compact subset of X, as 0 < a < IIfll. If E were a boundary for A, then E would be a closed boundary for A, and so E ~ oA. From this it would -

Q)

9.5. Applications of Boundaries

245

follow at once that a ~ inf ,f(s)I ~ inf If(s),, s€E s E3A

which combined with the preceding estimate would show that inf If (s) I

s EoA

=

inf

UfgllCD

g€A

II II . g

CD

g~O

Consequently it remains only to show that E is indeed a boundary for A. However, if E is not a boundary, then there exists some g E A, g ~ 0" such that suPsEElg{s)l < Ilg11 CD " as E is compact. Moreover" it is easily seen for each positive integer k that

sup s( E

ISk~S)1 = sup[lg(s)l]k, IIg II m sEE ~

from ""hich it follo\\'s at once that limk[suPsEE(lgk(s)l/llgkUCD)] = O. Furthermore" given k" there exists some tk E 3A such that If(tk)gk(t k)I = IIfgkU CD " whence we deduce that IIfgkUCD a <_r-"- \lgkHCD If{tk)gk(t k ) 1

< =

Thus

tk E E" k

= 1,2,3, •.. ,

Igk (t k ) I If (t k ) 1· and so

< IIfll sup CD 5 ( E

19k~s)l IIg IICD

(k = 1, 2 • 3, • • • ) •

9. Boundaries

246

But these estimates entail that Therefore E is a boundary for

a = 0, contrary to assumption. A, and the proof is complete.

o

The next corollaries are easily established with the aid of Theorem 1.6.2. The details are left to the reader. Corollary 9.S.1. Let X be a locally compact Hausdorff topological space and let A be a separating function algebra on X. If f E A, then the following are equivalent: (i)

f

is a topological zero divisor.

Corollary 9.5.2. Let X be a locally compact Hausdorff topological space and let A be a separating function algebra on X such that oA is compact. If f E A, then the following are equivalent: (i) (ii)

f

is a topological zero divisor. t E aA such that

There exists some

f(t)

= o.

In particular, Corollary 9.5.2 is valid whenever X is compact. It may not, however, be valid for noncompact X. For example, let X = lR and A = C OR). Then aA = lR. Let f E C (lR) be such that o -t 2 0 f(t) ~ 0, t E lEt For instance, let f(t) = e , t E IR. If gk E Co (lR) is the tent function defined for each positive integer k by gk(t)

=t

k< t < k

- k for

gk(t) = -t + k + 2 for gk(t) = 0 for

t

~

[k,k

+

1,

k + I < t < k +

+

2,

2],

then it is easily seen that IIgk llm = 1. k = 1.2.3, •••• and li~lIfgkllCD = o. Thus f is a topological zero divisor in A = Co (lR) • but f never vanishes on aA =lR.

9.5. Applications of Boundaries

247

In Section 1.6 we discussed some results about topological zero divisors similar to the preceding theorem and corollaries. In particular, the reader should compare the preceding results with Theorems 1.6.1 and 1.6.2. next corollary.

Theorem 1.6.1 is helpful in proving the

Corollary 9.5.3. Let X be a compact Hausdorff topological space and let A be a separating function algebra on X. If f E A, define ~(f) E C(oA) by ~(f)Ct) is the restriction of f to oAt

= f(t),

t (oA;

that is,

~(f)

Then

(i) The mapping ~: A - CCoA) morphism of A into C(oA).

is an isometric algebra iso-

(ii) If f (A, then ~(f) is regular in CCoA) if f is not a topological zero divisor in A.

if and only

If we shift our attention to commutative Banach algebras, the preceding results combined with Theorems 1.6.1 and 5.1.1 can be used to establish the next corollary. Corollary 9.5.4. Let A be a semisimple commutative Banach algebra with identity such that there exists a K > 0 for which IIxll 2 < Kllx 211, x E A. If x E A, then the following are equivalent: (i) (ii)

x is a topological zero divisor. There exists some T E oA

such that

•X(T) = O.

Our final application will be to obtain some results concerning the extension of maximal ideals. In particular, we shall show that every maximal ideal in the §ilov boundary of a closed subalgebra A of a commutative Banach algebra B with identity can be extended to a maximal ideal in B. More precisely, we have the following theorem: Theorem 9.5.3. Let B be a commutative Banach algebra with identity e and let A be a closed subalgebra of B that contains e. If M E oA c ~(A). then there exists some N E ~(B) such that N ~ M.

248

9. Boundaries

Let T denote the complex homomorphism of A such that M = T -1 (0) and suppose there exists no N E A(B) such that N ~ M = T-1(O). Then the ideal I in B generated by M must be all of Bj that is, the set Proof.

m

I = ( E XkYk 1 Xk € M, Yk E B, k = 1,2, .•. ,m),

k=l where m is an arbitrary positive integer, must be all of B. Otherwise, by Theorem 1.1.3, there would exist some N E A(B) for which N ~ I ~ M, contrary to assumption. In particular, there exist xk EM and Yk € S, k = 1,2, •.. ,n, such that e = tk=lx kyk , where, without loss of generality, we may assume that

lI~k IICD = sup

w£A(A) Furthermore, we observe that

I~k (w) 1 < 1

(k

= 1, 2, ... , n) •

sup 1~(UI)1 = limllxnll 1/n = sup l~(w)l wEArS) n w€A(A) by the Beurling-Gel'fand Theorem (Corollary 3.4.1). Now let

(x ( A),

a be so chosen that a>

=

•

sup [sup lYk(w)11 k=l,2, ... ,n w(A(B)

and consider the open neighborhood U of T in A(A), U = (u, = (w

defined by

• • Ixk(w) - Xk(T)1 < 1/2na, k = 1,2, ... ,n) • I Ixk(w)l

< l/2na, k

= l,2, ... ,n).

Since M = T-l(O) E oA, we see from Corollary 9.2.2 that there exists some yEA such that IIri! = 1 and I;(u.) I < 1/2na, w £ A(A) - U. Clearly y = ye = y(tk=lxkYk)' from which it follows that CI)

9.6. Representing Measures •

249

n •

•

ly(w) ~ xk(w)Yk(w)l = w~A(B) k=l sup

=

• lIyll •

= 1.

However, we also see that

•

n •

•

n...

sup ly(w) ~ xk(W)yk(w)l < sup [~ly(w)~k(w)lIYk(w)l] wEA(B) k=l w~A(B) k=l <

~ [sup ly(w)~k(w)l sup lyk(w)ll k=l wEA(B) wEA(B) n

••

•

= ~ [sup IY(w)xk(w)I sup lyk(w)ll k=l wE~(A) w(A(B) n

••

< a[ ~ sup ly(w)xk(w)ll k=l wEA(A) 1 < 2'

••

••

a: lr(w)~k(w)l.< IIY~.lJtk(w)l < 1/2na, W E u, k = 1,2, ... ,n, and ly(w)xk(w)l < IIxkll.1Y(w)1 < 1/2na, w € A(A) - u, k = 1,2, ... ,n. We have thus obtained a contradiction, and so there must be some N E A(B) such that N ~ M = T-I(O). o Theorem 9.2.3 immediately yields the next corollary. Corollary 9.S.S. Let B be a commutative Banach algebra with identity e and let A be a closed subalgebra of B that contains e. If A is either regular or self-adjoint, then for each M E A(A) there exists some N E A(B) such that N ~ M. It is not difficult to see that analogs of Theorem 9.5.3 and Corollary 9.5.5 exist for a commutative Banach algebra A with identity and superalgebras B of A. We leave the formulations of these results to the reader. Representing Measures and the Choquet Boundary. There are a number of equivalent descriptions of the Choquet boundary of a 9.6.

9. Boundaries

250

separating function algebra, which we shall discuss in this and the following section. One of these descriptions involves the notion of a representing measure, and we now wish to give some insight into the origin of this concept and at the same time indicate another fundamental connection between the theory of analytic and harmonic functions and the study of function algebras. With this is mind, suppose f E A(O) and CEO, lcl < 1. the Cauchy Integral Formula [LeRd, p. 133] asserts that 1

f{C) = 2ui Setting

z = e it ,-n

Ir

Then

fez) z _ C dz.

we see that this becomes it it f(C) = .!... f{~ dt 2n -n It , . e ~

t

~

u,

In

= Ir

fee

)e

it

) d~,(e

it

),

it it ) = ~ 1[2n(e - C)])dt. Thus we see that for each lc, < 1, there exists some complex-valued measure ~C in M(r) such that f(C) is obtained by integrating f on r with respect to ~C. Actually, for such C there even exist positive measures on r of norm one which have the same property. Indeed, if CEO, Ic' < 1, and z E r, then where , E OJ

d~~(e

it

z

zr

1

-z--~C = -(z---C-)~z-

= 1 - Cz

Moreover, it is apparent that h(z) = 1/(1 - Cz), z E 0, belongs to A(O), and so from the preceding observations we see that f(C)

-=~2 = (fb) (e) = 1 -

1,1

=

Ir

(fh. )

(e

1

= 2n

) ~e(e

it

)

it

.!... In 2n

it

f {e " ( _ _l---:~ d -n 1 _ eel't 1 _ Ce- l"t) t

In

-n

it

11

fee ~ dt. _ Ce- lt l 2

9.6. Representing Measures

251

lcl

< 1,

ICl 2

dt

Consequently, we see that for each C E D, f(C) = 1 r1T f(e it )

2n J_n

=

1 -

11 _ Ce-1t 12

we have

Ir fee it ) dve(e it ),

where dVeceit) = [(1 - 1'12)/2ncll - Ce- it I2 )]dt. is a positive measure in M(r) and IIvell = I r dVC(e

it

Evidently

) = 1.

. The funct10n P,(e it ) = (1 - Ie 12 )/2n( I 1 - Ce -it l 2 ), eit E r, of course. just the Poisson kernel [A, pp. 165-167]. Furthermore. we claim that in Mcr) such that

ve

ve

is,

is the only positive measure

Cf E A(D)). Indeed. suppose ~e E M(r) k = 0,1,2 •••• we have

Ir

e

ikt

is another such measure.

d(V, - ~C)(e

it

) =

k

C -,

k

=

Then for each

°

and

= 0,

as the measure Vc -~, is real valued. nometric polynomial on r, we see that

Ir

pee

it

) d(V, -

~,)(e

Thus, if p is any trigoit

) = 0,

whence it follows, by the Stone-Weierstrass Theorem [L, p. 332], that (h £

e(r»).

9. Boundaries

252 Hence we conclude~ either from the regularity of fact that the dual space of C(f) is M(f) , that

v, - A, or the v, - A, = o.

The preceding development can be summarized by saying that for each D, Icl < 1, there exists a unique positive measure

,E

\lC E: M(r) , IIvcll = 1,

such that f(C) =

Ir f(e it )

dV,(e it )

(f

Moreover, this result also holds for ,E D such that If C E r, then it is apparent that, if with unit mass concentrated at C, then fCC)

= Ir

fCe

it

Vc

= 6 C'

"t

) dV,(e 1 )

E A(D)).

1,1

= 1.

the measure

Cf E A(D)),

6,

and as before, is the only such positive measure with norm one. Indeed, suppose r and define g(z) = Cl + ,z)/2, z ~ D. It is easily verified that g E: A(D), IIgll CID = g(C) = I, and Ig(z)1 < 1, z £ D, z ~ C. Consequently it is evident that limkgkcc) = x{C)(z), z £ D, where xC,} denotes the characteristic function of the singleton set {C). Thus, if AC E M"(r) is any positive measure such that IlA,lI = 1 and for which

,E

(f £

A(D)) ,

then, by the Lebesgue Dominated Convergence Theorem [Ry, p. 229], we deduce that 1

= lim k

g k CC) =

"I

11m r gk (e it ) dACCe it ) k it

.t

= Ir x{C)(c ) dACce1 ) = AC({'))'

Thus we see that for each C £ D there exists a unique positive

9.6. Representing Measures measure

~,(

M(r)

253

of norm one such that f(,) =

Ir fee it )

dv,(e

it

)

(f £ A(D)).

Recalling the description of the maximal ideal space of A(D) given in Section 4.4, we see that the points of D correspond precisely to the complex homomorphisms of A(D). Thus, denoting by T, the element of ACA(D)) = D corresponding to " we see that there exists a unique positive ~,E Mcr) such that (f E A(D))

and for which

Moreover, we observe that in this case r = 3A{D). Hence in a rather obvious sense the measure ~, represents the complex homomorphism T"

which prompts the following general definition: Definition 9.6.1.

space and let the constants. then ~ E M(X) Ux*1I

= II~II

Let

X be a compact Hausdorff topological

A be a separating function algebra on X that contains If x* is a continuous linear functional on A, is said to be a representing measure for x· if

and x*(f) =

Ix

f(s) ~(s)

The reader should note carefully that, if x* E A* are such that x*(f)

= Ix

f(s) d~(s)

(f E A). and

~

E M(X)

(f E A),

then ~ is a representing measure for x* only if IIx*1I = II~II. If t E X and Tt denotes the complex homomorphism of A defined by Tt(f) = f(t), f £ A, then we shall frequently refer to a representing measure for Tt as a representing measure for t. Evidently ~ = 6t , the measure with unit mass concentrated at t, always is a represent-

9. Boundaries

254

ing measure for t £ X. However, it may not be the only such measure. 0, "I < 1, then on setting For example, if A = A(O) and dv,(e it ) = [(1 - I, 12 )/2n{ 11 - 'e -it l2 )]dt we see that v, E N{r) is a representing measure for , that is different from 6,. Representing measures for points are always positive. This is the content of the next proposition.

,£

Proposition 9.6.1. Let X be a eompact Hausdorff topological space and let A be a separating function algebra on X that contains the constants. If t E X and ~ E MCX) is a representing measure for t, then ~ ~ o. Proof. ving that

The assertion follows at once from Lemma 9.4.1 on obser-

Referring once again to the case of A = A(O) , we see that each point 0 has a unique representing measure -- namely, v, -such that the support of v, is a subset of the §ilov boundary r for ACD). Thus we are naturally led to the following questions about separating function algebras A on compact Hausdorff topological spaces that contain the constants: Which points in X have unique representing measures? Does there exist a representing measure ~ for t E X such that ~ E NcaA)? We shall see in Section 9.S that the answer to the second question is always in the affirmative, while the answer to the first question is that the points of X that have unique representing measures are precisely the points of the Choquet boundary for A.

,€

However, before considering these questions in detail, we wish to use the language of representing measures to prove an important result about the algebras Ccr) and A(O). We noted in Section 4.4 during our discussion of the maximal ideal space of ACD) that the algebra A(O) can be identified isometrically with a subalgebra B(f) of C(I). The mapping that effects this identification is, of

9.6. Representing Measures

255

course, just the mapping that associates with f E A(D) the restriction of f to f. Recall that 6(B(f)) = D. We now wish to prove that B(f) is a maximal subalgebra of C(f). More precisely, we have the following definLtion and theorem: Definition 9.6.2. Let X be a compact Hausdorff topological space and let A be a separating function algebra on X that contains the constants. If A is a closed subalgebra of C(X), then A is said to be a uniform algebra. It should be observed that a uniform algebra is a commutative 8anach algebra with identity. Theorem 9.6.1 (Wermer's Maximality Theorem). Let A be a uniform algebra on f. If 8(I) c A, then either A = B(f) or A = C(f). Proof. Since 8(f) c A, it is evident that the function h(e it ) = e it , e it E f, belongs to A. Suppose that T(h) ~ 0, T E 6(A). Then, by Corollary 3.4.2, h is regular in A, from which we conclude at once that the function -h(e it ) = h -1 (e it ) = e -it , e it E f, belongs to A. However, an easy application of the StoneWeierstrass Theorem [L, p. 332] shows that the algebra generated by hand h is dense in C(f), whence we deduce that A = Cef), as A is a closed subalgebra of C(f).

On the other hand, suppose there exists some T E 6(A) such that T(h) = o. Since 8(f) c A, it is apparent that T E 6(8(f)) = 6(A(D)) = D and. moreover, that T(f) = f(O), f E B(f). Note that the latter equation is meaningful, as we may identify f E Bef) with a unique element of A(D). The proof of the equation is as in Section 4.4. Let ~ E M(r) be a representing measure for T considered as an element of 6(A). From Proposition 9.6.1 we see that ~ > O. Moreover, we see that, for each f E B(f),

9. Boundaries

256

However, our development at the beginning of this section entails that ~ = Vo since the representing measures in M(r) for the points of 0 are unique. Thus we have f(O)

= T(f) = Ir

't it f(e 1 ) dvO(e ) (f € A).

In particular, if f E A,

then

o = hk(O)f(O) = T(hk)T(f)

= T(hkf) = ~Jn 2n -n

..

= f(-k) that is, However, that the f on r

eiktf(eit) dt (k

= I ~ 2 , 3, ..• ) ;

the negative Fourier coefficients of each f £ A are zero. a classical theorem of Fourier series [E 2 , p. 87] asserts Cesaro means of the Fourier series of a continuous function converge uniformly to f; that is,

o = limllan (f) n

= lim[ n

- fll CD

sup -n
Thus for each f E A we see that a (f) £ B(I), n = 0,1,2, ... , as .. n h E BCI) and f(-k) = 0, k = 1,2,3, ... , whence it follows that f € BCf), as B(r) is supremum norm closed. Therefore A = B(f), and the theorem is proved. o This proof of Wermer's Maximality Theorem is due to Hoffman and Singer. A proof, due to P.J. Cohen, that does not depend on the use of representing measures is also available. This proof as well as some further discussion of maximality theorems and their applications can be found in [Ga, pp. 38-40; S, pp. 297-299, 302-303, 340].

9.6. Representing Measures

257

Finally, we shall establish the description of the Choquet boundary indicated before Wermer's Maximality Theorem: a point t lies in the Choquet boundary if and only if it has a unique representing measure, which of course must be 6t , the unit mass concentrated at t. Theorem 9.6.2. Let X be a compact Hausdorff topological space and let A be a separating function algebra on X that contains the constants. If t E X, then the following are equivalent: (i)

(ii)

t

The representing measure for

Proof.

xA;

E XA.

t

is unique.

Before we begin the proof we recall the definition of

namely, if for each t E X we set

= f(t),

Tt(f)

f E A,

and

B~ = (x* 1 x* E A*, IIx*1I = x*(l) ~ 1), then xA

= (t 1 t E x,

Tt E ext(B~)),

where, as usual, ext(B~) denotes the set of extreme points of For further details the reader is referred back to Section 9.4. Suppose

t E XA.

measure for t, then ~ > 0,

then

~

We must show that, if ~

= 6t . II~ II

However, if

= JX <4J, (s )

~

B~.

is a representing

is such a measure,

= I,

and f(t)

= Ix

f(s) ~(s)

Moreover, if E C X is any Borel set such that xi and x2 defined by xi(f) xi(f)

1 = ~(E) SE

(f

0<

~(E)

< 1,

E A). then

f(s) ~(s),

= ~(X:E) I X-

E f(s) <4J,(s)

(f

£ A),

258

9. Boundaries

are elements of B~ f(t)

such that

= Tt(f) = ~(E)xi(f)

Since Tt E ext(B1l ),

+

[1 -

~(E)]xi(f)

(f E A).

we conclude that

Tt(f) =

Ix f(s)

= ~~E)fE =

~(s)

f(s)

~(X:'E) I X-

~(s) E f(s)

d~(s)

(f E A).

Then, appealing to Lemma 9.4.2 and arguing precisely as in the proof of Theorem 9.4.1, we see that the support of ~ must be a single point. But since f(t)

= Ix

f(s) ~(s)

the support of ~ must be the singleton set

(f

ttl,

E A),

that is ~ = 6t .

even simpler argument gives the desired conclusion in the case that for each Borel set E C X we have ~(E) = 1. Thus part (i) of the theorem implies part (ii). An

Conversely, suppose t has a unique representing measure, which of course must be 6t • We need to show that Tt E ext(B~). If xi,xi E B~ and 0 < a < I are such that Tt = axi + {I - a)xi' and if ~l and ~2 are representing measures for xi and xi, respectively, we see that 6t = ~l + (1 - a)~2 since -"1 + (1 - a)~2 is a representing measure for t, and the representing measure for t is unique. However, as mentioned in the proof of Theorem 9.4.1, the measure 6t is an extreme point of the closed unit ball in M(X), whence we conclude that 6t = ~l = ~2· Consequently, Tt = xi = xi and Tt E ext{B 1I ); that is, t E XA. 0 In view of our discussion of A(D) at the beginning of Section 9.6, we see at once from the preceding theorem that the Choquet boundary of A(D) is precisely f. Thus, since the Bishop boundary of

9.7. Characterizations of the Choquet Boundary

259

A{D) exists and is also equal to r, we have pA{D) = XA{D). We shall see at the end of the following section that this is always the case for uniform algebras on compact metric spaces. 9.7. Characterizations of the Choguet Boundary. The main purpose of this section is to establish a number of characterizations of the Choquet boundary for uniform algebras. The characterizations will consist of four necessary and sufficient conditions for a point to lie in the Choquet boundary. Before we can state and prove the theorem we need a number of preliminary results and definitions. Definition 9.7.1. Let X be a compact Hausdorff topological space. Then CR(X) will denote the Banach space over lR consisting of all the real-valued continuous functions on X with the supremum norm. A linear functional F on CR{X) is said to be positive if R F{f) > 0 whenever f E c (X) and f{t) ~ 0, t E X. Lemma 9.7.1. Let X be a compact Hausdorff topological space. If F is a positive linear functional on CR(X), then F is continuous and IIFII = F{l). Proof. Since F is positive, it is apparent that, if f and g belong to CR(X) and f(t) ~ get), t E X, then F(f) > F(g). Thus, if f E CR{X) is such that IIfllCD < 1, then it follows easily that -F(l) ~ F(f) ~ F(l), whence IF(f), < F(l). Consequently F is continuous and UFII = F(l). o In the statement and proof of the next lemma we shall use r(x*) to denote the set of representing measures for a given continuous linear functional x*. Lemma 9.7.2. Let X be a compact Hausdorff topological space, let A be a uniform algebra on X, and let T E 6(A). For each h € CR(X) define ~(T,h) = sup{Re[T(f)]

I

f ( A, Re[f(s)] < h(s), s E

xl

9. Boundaries

260

and Q(T,h) If

~

(r(T),

= inf{Re[T(f)] I

f £ A, Re[f(s)] ~ h(s), s (

xl.

then Q(T,h) <

Ix

h{s) d~{s) < Q(T,h)

and if h (eR(x) and a (R are such that then there exists some ~ E reT) for which a

= Ix

Q(T,h) < a < Q(T,h),

h(s) d~(s).

Moreover, for each h ( eR(x), ~(T.h) = inf{Ix h(s) d~(s)

I~

E reT)}

and Q(T.h)

= sup{!x

h(s) d~(s)

I

~ E reT)}.

Proof. Suppose h (CR(X). If ~ E reT), such that Re[f(s)] ~ h(s), sEX, we have Re[T(f)]

= Re[Ix f(s) = Ix <

since

~

> O.

Ix

then for each

f EA

d~(s)]

Re[f(s)] ~(s) h(s) d~(s),

Hence ~(T,h) < Ix h(s) d~(s)

(~

( r(T».

(~

E r(T»,

Similarly we deduce that Ix h(s) d~(s) < Q(T,h) which proves the first assertion of the lemma. Next we note that Q(T,.)

is a positive homogeneous subadditive

9.7. Characterizations of the Choquet Boundary

function from

CReX)

to ~;

261

that is,

Q(T,bh) = bQ(T,h), Q(T,h

+

g) < Q(T,h)

+

R

(g,h E C (X); b > 0).

Q(T,g)

We shall need this easily verified observation in a moment. Now let h E CRex) and suppose a En is such that Q(T,h) < a < Q(T,h). Set W= (bh I b E IR). Evidently W is a linear subspace of CR(X). We define a linear functional F on W by F (bh) = ba, b E~ o 0 From the restrictions on a we see at once that F (bh) = ba < Q{T,bh), o b E kL Thus, since QeT,·) is a positive homogeneous subadditive function on eReX), we may appeal to the Hahn-Banach Theorem for real linear spaces [L, p. 83]to deduce the existence of a linear functional F on CR(X) such that F restricted to W is equal to Fo and such that Feg) ~ -QeT,g), g E CR(X). In particular, if g (CR(X), then Fe-g) ~ Q(T,-g), whence F(g)

= -Fe-g)

> -Q(T,g)

= Q(T,g).

Moreover, an easy computation using the definition of that, if g E CReX) and g(s) > 0, sEX, then

o = ~eT,O) <

~(T,g) ~

~(TJh)

reveals

peg),

where 0 is used to denote both the number zero and the function in CRex) that is identically zero. Hence F is a positive linear functional on eRex), and so, by Lemma 9.7.1, F is a positive continuous linear functional on CR(X) and IIFIi = Fel). Appealing to the Riesz Representation Theorem for such functionals [L, p. 109], we deduce the existence of some ~ E M(X), ~ ~ 0, such that peg) = Ix g(s) d~(s) and for which 1l~1I =

Ix ~(s).

9. Boundaries

262

Furthermore, if f E A, then Re[f] (CR(X) and ~(T,Re[f]) Q(T,Re[f]) = Re[T(f)]. The details of the latter assertions are left to the reader. Hence we see that

=

= F(Re[f])

Re[T(f)]

=

Ix Re[f(s)]

d~(s)

=

Re[ix f(s)

d~(s)]

(f E: A),

from which it follows at once that T(f) Moreover, since T(f)

• = f(T),

11TH = and so

~

E reT).

= Ix

f(s) ~(s)

(f ( A) •

we see that

1 = T (1) =

I x ~ (s)

= II~II,

Finally we observe that

Ix h(s)

~(s)

= F(h) = Fo(h) = a,

and the proof of the second assertion of the lemma is complete. The last assertion of the lemma is now apparent.

o

One more definition is necessary before we can turn to the main theorem. Definition 9.7.2. Let X be a locally compact Hausdorff topological space and let A be a separating function algebra on X. A subset E C X is said to be a peak set for A if there exists some f E A such that E = (t I t E X, If(t)1 = IIflle] = Ef . A point t E X is said to be a weak peak point for A if there exists a family of peak sets (Ef] for A such that n Ef = (t). A a a a point t E X is said to be a strong boundary point for A if for each open neighborhood U of t there exists some f E A such that

II f lie = If (t) I = I

and

1f (s) 1 <

I, sEX - U.

9.7. Characterizations of the Choquet Boundary

263

Clearly a peak set Ef is a nonempty compact subset of X provided f is not identically zero. The definition of a strong boundary point should be compared with the characterization of points in the §ilov boundary given in Corollary 9.2.2. Our main result is the following theorem: Theorem 9.7.1.

Let

X be a compact Hausdorff topological space

and let A be a uniform algebra on X. ing are equivalent:

If t

E X, then the follow-

(i) Given 0 < a < a < 1, if U is an open neighborhood of t, then there is some f E A such that IIfll., < 1, If(t)l > a, and If(s)I < a, sEX - U. (ii)

If U is an open neighborhood of t,

some f E A such that sEX - U.

IIfU., ~ 1, lfet)1 > 3/4,

(iii)

t

is a strong boundary point for

(iv)

t

is a weak peak point for

(v)

t

then there exists and

If(s)l < 1/4,

A.

A.

E xA.

Proof. Obviously part (i) implies part (ii), so we assume that part (ii) holds and let U be an open neighborhood of t. We must construct some g E A such that Iget)1 = IIgll., = 1 and Ig(s)1 < 1, sEX - U. To accomplish this we shall first define a sequence of open neighborhoods {Uk] of t and a sequence {gk] C A such that for each k we have (a)

Uk ::> Uk+ 1 ·

(b)

gk (t) = 1.

(c)

1I1)t1l., <

Cd)

Ilk(S)' < 1/3, sEX - Uk.

4/3.

9. Boundaries

264 (e)

sup

s

Ig·(s)1 < 1 + 1/(3.2k), j = I,2, ... ,k - l.

£U ~

k

J

-

UI = U. Then there exists some fl E A such that IIfIIICD < 1, Ifl(t)1 > 3/4, and Ifl(s)1 < 1/4, sEX - UI . Moreover, we may assume without loss of generality that we even have fI(t) > 3/~, since if this is not true of f l' it certainly is valid for e -1 arg f 1 (t) fl' The other The definition of these sequences will be inductive.

properties of fl

Set

are clearly unchanged by this convention.

Now

define gl = fI/fI(t). Clearly gl E A, gl(t) = 1, /lgIlioo < 4/3, and Igl(s)1 < 1/3. sEX - Ul . Conditions (a) through (e) are vacuously satisfied for gl' Suppose that UI ,U 2""'Un and gl,g2, ... ,gn

(e).

have been defined and satisfy conditions (a) through

Then define U +1 n

Un+ 1 as

=

n

n (s

k =1

I

n

1

sEn U., Igk(s)1 < 1 + j =1 J

3 • 2n +

1 ).

Since t E n~=IUj and gj(t) = 1, j = I,2, ... ,n, it is apparent that Un+l is an open neighborhood of t and Un +l C Un. Let fn+ I E A be such that IIfn+ 111CD -< 1, f n+ I(t) > 3/4, and Ifn+1 (s)1 < 1/4, sEX - Un+1 , and define gn+I = fn+I/fn+I(t). As before, it is evident that gn+I E A satisfies conditions (b) through (d). Moreover, if s E Un+ l' then from the definition of U it follows that n+l (k =

1,2, ... , n) .

Proceeding in this way, we define inductively a sequence of open neighborhoods (Uk) of t and a sequence (gk) C A that satisfy conditions (a) through (e). Now define g as g = Ik=I gk/ 2k . Since A is a uniform algebra, we see that g E A. Furthermore, since gk(t) = 1, k = 1,2,3, ... , it is obvious that get) = 1. If sEX - U then sEX - Uk' k = 1,2,3, ... , and so J

9.7. Characterizations of the Choquet Boundary Finally, suppose

s E U.

If

s E

CD

~=IUk'

265

then for each

j

Ig·(s)1 < sup 19.(s)1 J -sEU J k

< I + 1

(k

3":2k

whence we conclude that s E s ~ s ~ (c)

=j

Ig.(s)1 < 1, j = 1,2, •..• J

+

1,j + 2, ... ),

Thus, if

-

then Ig(s)1 < 1. On the other hand, if s E U but ~=lUk' then there exists some m ~ I such that s E: Um but U +1. Then sEX - U ., j = 1,2,3, •.. , and so conditions m m+J through (e) combined entail that

Ok=lUk'

m-l Igk(s)1 Ig(s)1 < ~ k=l 2k

<

I

Igm(s)1 +

m-1 I

(1 + - ) 1: -

3.2m k=l 2k

2m

+ -

CD

+

t

3.2m

k 2

k=m+1 I

4

Igk(s)l

+ -

CD

~

-

1

3 k=m+l 2k

4 3.2m

1 3·2m

+-+--

=1 __1.....",..._ 3.2 2m - 1 <1.

E A is such that 11&11 = get) = 1 and 19(s) I <

Thus

g

5 ~

X - U,

for

CD

and so we conclude that

t

1,

is a strong boundary point

A. Next suppose that t is a strong boundary point for A and let (U) be a family of open neighborhoods of t such that an0'0' U = {t}. Since t is a strong boundary point, we see that for each U there exists some f E A such that IIf II = If (t) I = 1 a a rteD a and If (s)1 < 1, sEX - U. Setting Ef = {s I sex, IIf II = If (s)ll, a a aam awe see that t E Ef C U, that Ef is a peak set, and n Ef = (t). a a a a a Hence t is a weak peak point.

266

9. Boundaries

Now suppose t is a weak peak point and let ~ be a representing measure for t. If U is an open neighborhood of t, then, since t is a weak peak point, there exist some f E A and a closed set E c: X such that t E: E C U and E = (s I sEX, IIfU = If(s) 11 • Without loss of generality we may assume that 1 = IIfU = f(t) • Then we see at once that

..

o=

..

1 - f(t)

Ix [1 = Ix Re[l =

f(s)] ~(s) - f(s)] ~(s),

as ~ > 0 by Proposition 9.6.1. But since If I assumes its maximum precisely on E~ we deduce that the nonnegative function Re[l - f) cannot be identically zero on X - U. Thus ~(X - U) = o. Since U is an arbitrary open neighborhood of t and ~ is regular, it follows at once that the support of ~ is the singleton set (t), and so ~ = Qt. Hence the representing measure for t is unique, and t E XA by Theorem 9.6.2. Finally, suppose that t E XA and 0 < a < ~ < 1. If U is an open neighborhood of t, then, using Urysohn's Lemma [W2 ' p. 55] it is easy to see that there exists some h E CR(X) such that h(t) = 0, h(s) < log a, sEX - U, and h(s) ~ 0, sEX. Consequently, since the only representing measure for t is ~ = 0t by Theorem 9.6.2, we see from Lemma 9.7.2 that

o = h(t) = Ix

h(s) dOtes)

= inf(Ix h(s)

~(s) I ~ E r(Ttl]

= Q,(T ,h) =

sup{Re[Tt(f)]

=

sup(Re[f(t)]

I

I

f E A, Re[f(s)] ~ h(s). s E

xl

f E A, Re[f(s)] ~ h(s), s E

xl.

9.7. Characterizations of the Choquet Boundary

267

Hence, since log ~ < 0, we conclude that there exists some g ~ A such that Re[g(s)] < h(s), sEX, and Re[g(t)] > logS. Let f = ego Since A is a uniform algebra, it is evident that f E A, and some elementary computations reveal that IIfli. ~ I, If(t) I > (:), and If(s)1 < a, sEX - U. This establishes the equivalence of all five parts of the theorem and completes the proof.

o

Theorem 9.7.1, combined with Theorems 9.6.2 and 9.4.1, gives a rather complete description of the Choquet boundary of a uniform algebra. Moreover, the results allow us to show quite easily that the Bishop boundary exists for any uniform algebra A on a compact metric space. First we require a lemma. Lemma 9.7.3. Let X be a compact Hausdorff topological space and let A be a uniform algebra on X. If t E X, then the following are equivalent: (i)

t

is a peak point for A.

(ii)

t

is a strong boundary point for A and

Proof. If boundary point. IIfll = If(t)1 = • n let Un = {s open, t E Un'

it)

is a G6-set.

t is a peak point, then it is clearly a strong Moreover, there exists some f E A such that 1 and If(s)1 < 1, s ~ t. For each positive integer 1 sEX, If(s)1 > 1 - lIn). Evidently each Un is and n:=IUn = ttl. Thus ttl is a G6-set.

Conversely, suppose t is a strong boundary point and ttl is a G6-set. Then there exists a decreasing sequence {Un) of open neighborhoods of t such that n:=lUn = ttl. For each Un there is some f n E A such that IIfnII . = f n (t) = 1 and Ifn (s)1 < 1, s ~ X - U. On replacing f by a suitable power of itself, we may n n assume without loss of generality that If (5)l < 1/2, sEX - U • n n n Let g = ~=lfn/2. Clearly g E A and get) = 1. If sEX and

9. Boundaries

268 and s; t, then either s ~ that s ( U and s ~ Um+ 1. m Ig(s)1 < 1/2, whereas in the

U1 or there exists some m > 1 such In the first instance we have second we see that

mIl 1 < 1: + E n=l 2n 2 n=m+l 2n CX)

I

= I --

2m+ 1

<1.

Hence t

is a peak point for A.

o

Theorem 9.7.2. Let X be a compact Hausdorff topological space and let A be a uniform algebra on X. If X is metrizable, then pA = xA. Proof. Since X is metrizable, it is obvious that every point of X is a Go-set, and so by Lc~ma 9.7.3 a point t E X is a peak point if and only if it is a strong boundary point. However, by Theorem 9.7.1, the Choquet boundary, which always exists, consists precisely of the strong boundary points, whence pA = XA. o As already noted in Example 9.3.5, if X is not metrizable, then the Bishop boundary may fail to exist. Using Theorem 9.7.ICi) or Cii), one can show, when X is compact and metric, that the Choquet boundary of a uniform algebra A on X is a Go-set. As indicated in the remarks following Corollary 9.4.1, this may not be the case if X is not metrizable. We state this result as the next corollary, whose proof is left to the reader.

9.8. Representing Measures and the ~ilov Boundary

269

Corollary 9.7.1. Let X be a compact Hausdorff topological space and let A be a uniform algebra on X. If X is metrizable, then

pA = XA is a G6 -set. 9.8.

Representing Measures and the ~ilov Boundary.

,E

Returning

again to the algebra A(D), we recall that, if int(D), there exists a representing measure ~, for , such that

v,

then E M(l)

= M[aA(D)].

The next theorem shows that an analogous result is valid for any separating function algebra on a compact Hausdorff topological space that contains the constants. Theorem 9.8.1. Let X be a compact Hausdorff topological space and let A be a separating function algebra on X that contains the constants. If t E X, then there exists a representing measure ~ for t such that ~ E M(oA).

of of is is

Proof. Let B(aA) denote the subalgebra of C(aA) consisting the restrictions t6 aA of the elements in A. From the nature the ~ilov boundary it is apparent that the restriction mapping an isometric algebra isomorphism of A onto B(aA) and that aA a compact Hausdorff topological space. We define a continuous

linear functional 1t on B(aA) by Tt(fl oA ) = f(t), f E A, where flaA denotes the restriction of f to oA. Clearly IITtll = Tt(llaA) = 1. By a consequence of the Hahn-Banach Theorem [L, p. 87] we see that Tt can be extended to a continuous linear functional on C(oA) without increasing the norm, and so, by the Riesz Representation Theorem [L, p. 109], there exists some ~ E M(oA) such that (f E A).

Obviously complete.

~

is a representing measure for

t,

and the proof is

o

This result leads to the following corollary about commutative Banach algebras:

9. Boundaries

270

Corollary 9.8.1. Let A be a commutative Banach algebra. T E ~(A), then there exists some ~ E MCoA) such that Ci) (ii) (iii)

~

o.

>

:5.

~ (oA)

T(X)

Proof.

If

1.

• • = X(T) =foA lew)

~(w), x

E A.

If A has an identity, then the conclusion follows at

from Theorem 9.8.1 applied to A = •A. Suppose A is without identityand consider the algebra A[e]. From Theorems 3.2.1 and 3.2.2 we recall that ~(A[e]) = ~(A) U (T J. We claim that Q)

oA c oA[e] c oA U {T

Q)

Indeed, since the embedding of A{A)

into

J. A(A[e])

is a homeomor-

phism and oA is closed, we see that oA can be considered as a closed subset of A(A[e]) that is the minimal closed boundary for • • • A C A[e] . Thus oA c oA[e]. Since the functions ae obviously attain their maximum absolute value only on {T), we conclude that CXI oA[e] c oA U {TCDJ. Applying Theorem 9.8.1 to A = A[e] , we deduce the existence of some ~ E M(oA[e]) such that ~ > 0, ~(oA[e]) = 1, and for which

•

X(Te)

where, as usual,

T

A[e] determined by restricted to oA. ~ - V{{T )6 T . In CD CD so, in either case, M(oA) such that ~

e

• = foA[e] x{w)

d~(w)

(x E A),

denotes the unique complex homomorphism on

T. If ~({T) = 0, then we let ~ be v CD while if ~({TCDJ) ~ 0, then we set ~ = this case it is apparent that ~((T)) = 0, and CD ~ restricted to oA defines an element of > 0, ~(oA) :5. I, and for which T(X)

• = •X{T) = faA x(w)

~(w)

(x E A).o

9.S. Representing Measures and the §ilov Boundary

271

The representing measure given by Theorem 9.8.1, of course, need not be unique. Indeed, in view of Theorem 9.6.2, it will be unique precisely when t belongs to the Choquet boundary.

An analog of Theorem 9.S.l with the §ilov boundary replaced by the Choquet boundary is also valid. Its proof, however, depends on a deep result, generally referred to as the Choquet-Bishop-deLeeuw Theorem, which could not be fully presented without a considerable digression. Consequently we shall content ourselves with a statement of the theorem and refer the interested reader to [Ph; S, pp. 5S and 59] for further details. Theorem 9.S.2 (Choquet-Bishop-deLeeuw). Let X be a compact Hausdorff topological space and let A be a uniform algebra on X. If t E X, then there exists a representing measure ~ E M(X) for t such that ~(E) = 0 whenever E ~ X is a Baire set or a G6-set that is disjoint form xA. Extensive discussions of the use of representing measures in the study of polynomial and rational approximation as well as the significance of many of the other concepts introduced in this chapter for the investigation of function algebras can be found in [B, Ga, Lb, S, wm l , WID 2]. However, we shall not pursue the matter further since it would take us too far afield from the general theme of this volume.

CHAPTER 10 B*-ALGEBRAS 10.0. Introduction. A B*-algebra is a Banach algebra on which there exists a certain antilinear idempotent mapping called an involution such that a particular norm identity is valid with respect to this mapping. Such algebras are, in general, rather tractable objects of study. For example, we shall prove below that, if A is a commutative B*-algebra. then the Gel'fand transformation on A is an isometry of A onto C (A(A» that preserves the involution on A. o As we shall see shortly, the involution on C (A(A» is just complex o conjugation. This result, called the Commutative Gel'fand-Naimark Theorem, should obviously be a valuable tool in investigating commutative B*-algebras, and we shall devote the final five sections of this chapter to examining its application to some specific B*-algebras. In Section 10.3 we shall concentrate our attention on the commutative Banach algebra Ao(G) obtained as the closure in L(L 2 CG» of the continuous linear transformations Tf ~ L(L 2 (G», f E LICG), defined by TfCg) = f * g, g (L 2 (G). As usual, G is a locally compact Abelian topological group. The results we shall obtain in Section 10.3 about A (G), which is a B*-algebra, will enable us o to finally prove Plancherel's Theorem in the succeeding section. In Section 10.5 we shall establish the Pontryagin Duality Theorem, which depends directly on the regularity of LICG) and hence indirectly on Plancherel's Theorem. This theorem asserts, as indicated in • • is topologically isomorphic to G. Section 4.7, that CG) In the last two sections we shall utilize the Commutative Gel'fand-Naimark Theorem to obtain a spectral decomposition theorem for

272

10.1. B*-Algebras

273

self-adjoint continuous linear transformations on Hilbert spaces and to study the commutative Banach algebra of almost periodic functions on a locally compact Abelian topological group called Bohr compactification of G. 10.1.

B*-Algebras.

G and the so-

In this section we shall define algebras

with involution and B*-algebras, give a number of examples, and establish some elementary properties of B*-algebras. eral assumptions of commutativity. Definition 10.1.1.

Let

We make no gen-

A be a Banach algebra.

Then

A is

said to be a Banach algebra with involution if there exists a mapping * : A - A such that for any x,y E A and a (~ we have (i) (ii) (iii) (iv)

(x + y)* (ax)*

= x*

+

y*.

= ax*.

(xy) * = y*x*. (x*)*

= x** = x.

A Banach algebra A with involution is said to be a B*-algebra if IIx*xll = IIx1l 2 , x ( A. It is apparent that, if A is a commutative Banach algebra that is self-adjoint and semisimple, then A is a Banach algebra with involution. Indeed, in this case it happens that x € A for .. each x E A. If Y is the unique element in A such that y = x, then one sets x* = y. It is easily verified that the mapping * : A - A is an involution. Thus, for example, if X is a locally compact Hausdorff topological space, then c (X) is a Banach algeo bra with involution, where for f € Co(X) we set f*(t) = f(t), t E X; and if G is a locally compact Abelian topological group, then LI(G) is a Banach algebra with involution provided we define f* almost everywhere by the formula f*(t) = f(-t), t € G. Obviously Co(X) is also a B*-algebra. However, LI (G) is not generally

-- ....

274

10. B*-Algebras

a B*-algebra. Although we shall not prove this in detail, it is an immediate consequence of the Commutative Gel1fand-Naimark Theorem (Theorem 10.2.1) and the fact that LI(G) • is properly contained • unless G is a finite group [Ru , p. 90]. in Co (G) l On the other hand, there exist commutative Banach algebras with

involution that are not self-adjoint. One such algebra is A(D). It is easily seen that this is an algebra with involution provided we set f*(I) = f(Z), zED, for each f E A(D). It is, however, not self-adjoint, since, if f E A(D), then f E A{D) only if f is a constant, where of course fez) = fez), zED. Recall that the Gel'fand transformation on A(D) is just the identity mapping. That A(D) is not a B*-algebra will be immediately apparent from the Commutative Gel'fand-Naimark Theorem.

An important example of a B*-algebra is the Banach algebra A = L(V) of all the continuous linear transformations from a Hilbert space V over C to itself. As indicated in Example 1.2.11, L(V) is a Banach algebra with identity. It is a Banach algebra with involution since the mapping * : LcY) ~ L(V) defined by T ~ T*, T E L(V), where T* denotes the Hilbert space adjoint of T, satisfies the requirements of an involution as set forth in Definition 10.1.1. Moreover, L(V) is a B*-algebra. Indeed, if T E LcY), then IITII = IIT*II fL, p. 96], and so, on the one hand, we see that IIT*TII < IIT*III1TII ~ IITII2, while on the other hand, by a consequence of the Hahn-Banach Theorem [L, p. 89] and the Riesz Representation Theorem for Hilbert spaces [L, p. 390], we have IIT*TII = sup IIT*T(x)1I IIxll=1

= ~

sup [ sup I(T*T(x),y)l] IIxll=1 lIyll=1

sup I(T*T(x),x>1 - IIxll=1

10.1. B*-Algebras

275 = sup I
where <.,.) denotes the inner product in V. Hence IIT*TII = IITII2, T E L(V). Furthermore, it is evident that, if B is a norm-closed subalgebra of L(V) with the property that T E B implies T* E B, then B is also a B*-algebra. Such a subalgebra of Lev) is generally called a C*-algebra. One more definition is needed before we can prove some elementary properties of B*-algebras. Definition 10.1.2. Let A be a Banach algebra with involution. A subset E C A is said to be symmetric if x E E implies x* E E. An element x E A is said to be self-adjoint if x* = x. Thus, if V is a Hilbert space over ~, then those T E L(V) that are self-adjoint as elements of the B*-algebra L(V) are precisely the elements of L(V) that are self-adjoint as continuous linear transformations. A C*-algebra is bbviously a norm-closed symmetric subalgebra of L(V) for some Hilbert space V over C. Proposition 10.1.1.

Let A be a B*-algebra and let x E A.

Then (i)

(ii)

0*

= O.

If A has an identity e,

then

e*

= e.

(iii) If A has an identity e and x E A is regular, then x* is regular and (x*) -1 = (x -1 )*. (iv) If x E A is quasi-regular, then x* and (x*) -1 = (x -1 )*.

is quasi-regular

10. B*-Algebras

276 (v) (vi) (vii) (viii)

'~a(x)

if and only if

,E

a(x*).

IIx*1I = IIxli. If x is self-adjoint, then If x is self-adjoint, then a(x)

cr~

Proof. The proofs of parts (i) through (v) follow almost immediately from the mapping properties of the involution. The details are left to the reader. Part (vi) is evident if x = 0, and if x ~ 0, then lixll 2 = IIx*xll < IIx*lIl1xll, whence IIxll < IIx*U. Consequently IIx*1I < IIx**U = IIxll,

and so

IIx*1I

= IIxU.

From the Spectral Radius Formula (Theorem 2.2.2) we know that IIxlia = limnllxnUl/n for any x ~ A. If x is self-adjoint, then IIx 211 = IIx*xll = IIxU 2, from which it follows easily that IIx 2n ll = IIxll2n, n = 1,2,3, •.•• Thus n In proving part (viii) of the proposition we may assume, without loss of generality, that A has an identity e. This is possin

n

ble because, by Theorem 2.1.1, aA(x) = aA[e] (x). Suppose a + ib belongs to a(x) and let pt(x) = x + ite and qt(x) = x - ite, t Ent Evidently a + i(b + t) (a[Pt(x)], whence by part (v) we see that a - i(b + t) ~ a[Pt(x)*] = a[qt(x)], as x is self-adjoint. Thus by the Polynomial Spectral Mapping Theorem (Theorem 2.2.1) we deduce that a 2 + (b + t) 2 (a[ptqt(x)] = a(x 2 + t 2e). Hence from Theorem 2.1.2 we see that 2222 a2 + (b + t) = a + b + 2bt + t

< IIx 2 < IIx 211

that is,

a2

+

b2

+

t 2ell

+

2bt < IIx 211, t E If\.

+

t 2j

10.2. The Gel'fand-Naimark Theorem

277

This estimate, however, obviously leads to a contradiction unless b = o. Therefore a(x) c IR.

o

10.2. The Gel'fand-Naimark Theorem. We shall now state and prove the commutative version of the Gel'fand-Naimark Theorem, which asserts that the Gel'fand transformation on a commutative B*-algebra A is an isometric isomorphism of A onto C (A(A)) that preserves o the involution. We shall also mention briefly the noncommutative version of this theorem. We choose not to discuss in detail the noncommutative theorem since our applications of the Gel'fand-Naimark Theorem in the succeeding sections will all be in a commutative context. One more definition is required. Definition 10.2.1. Let A and B be Banach algebras with involutions * and +, respectively. A homomorphism ~ : A - B is said to be a *-homomorphism if ~(x*) = ~(x)+, x E A. If X is a locally compact Hausdorff topological space, then we shall consider C (X) as a Banach algebra with involution, as o discussed in the preceding section; that is, the involution on Co(X) is given by complex conjugation. Theorem 10.2.1 (Commutative Ge1'fand-Naimark Theorem). Let A be a commuta'ive B*-algebra. Then the Gel'fand transformation on A is an isometric *-isomorphism of A onto C (A(A)). o

Proof. If x E A is self-adjoint, then from Proposition 10.1.1 (vi) and the Beurling-Gel'fand Theorem (Corollary 3.4.1) we see that • IIxll = IIxli. Now, if x is any nonzero element of A, then clearly ~ y = x*x is self-adjoint and so Ilyll = lIyll. Thus by Proposition 10.1.I(vi) and the fact that A is a B*-algebra we see that

.

~

IIxll 2 = IIx*xll = lIyll •

I:

lIyll~

10. B*-Algebras

278

= lI(x*)·~1I < lI(x*) -

CD

• II IIxli • CD

CD

•

< IIx*lIl1xll -

=

CD

•

IIxllllxll , CD

•

whence we have IIxll < IIxli. Since the reverse inequality is always valid, we conclude that IIxli = IIxll , x E A. -

CD

•

CD

Hence the Gel'fand transformation is an isometric isomorphism • and, as always, A• separates the points of ~(A) of A onto A, and ZeAl = (T IT' A(A), X(T) • 0, X , AJ • ,. Moreover, if x E A, then x = y + iz, where y,z E A are self-adjoint. Indeed, y = (x + x*)/2 and z = (x - x*)/2i. Consequently from the properties of the involution * and Proposition 10.1.I(viii) we deduce that

..

..

(x*) • =

[(y +

iz)*] • = •y

iz•

= •y

+

•

.-

iz = x.

.

Thus we see that the Gel'fand transformation is an *-isomorphism and that A is closed under complex conjugation. Therefore by the .Stone-Weierstrass Theorem [L, p. 332], we conclude that A = C (~(A», o which completes the proof.

o

Corollary 10.2.1. Let A ~e a commutative B*-algebra. Then A is a Tauberian, self-adjoint, semisimple, regular, commutative Banach algebra. The problem of spectral synthesis is also quite easily solved in commutative B*-algebras, as seen from the next corollary. Corollary 10.2.2. (i) If E C synthesis. (ii)

If

I

~(A)

C A

Let A be a commutative B*-algebra.

Then

is closed, then E is a set of spectral

is a closed ideal, then

I

= k[h(I)].

10.2. The Gel'fand-Naimark Theorem

279

Proof. In view of our discussion of spectral synthesis in Section 8.3, it is apparent that we need only prove part (ii). Moreover, to establish part (ii) it suffices, by Proposition 7.1.I(vii) and Theorem 8.1.I(ii), to show that I o [hell] is dense in k[h(l)] since I [h(I)] C I C k[h(I)] and I is closed. Recall that o • I [h(I)] consists of all those x (A such that x vanishes idento icallyon some open set Ox ~ h(l). With this in mind, let y 'k[h(I)] and let e > O. Obviously the set U

= {T I

• T E ~(A), (y(T)1 < e)

is an open subset of ~(A) such that U ~ h(I), and X - U is compact. For each T E ~(A) let yeT) = p(T)e i9 (T), where peT) > 0 and 0 S 9CT) < 2n, and define f on ~(A) by f(T) = 0 for

T E U,

f(T) = [peT) - e]e i9 (T)

for

T E X-

u.

It is easily verified that f E Cc (~CA)) C C0 (~(A)). Hence, by the Commutative Gel'fand-Naimark Theorem, there exists some x E A such • that x = f. Evidently x E I [h(I)], and again appealing to the o • • preceding theorem, we see that lIy - xII = lIy - XIlCD ~ e. Hence lo[h(I)] is dense in k[h(I)], and so I = k[h(I)]. o Thus we see that the closed ideals in a commutative B*-algebra are precisely the sets I of the form I = keEl for some closed set E C ~(A). These observations provide a new proof of this fact for the B*-algebra C(X), X being a compact Hausdorff topological space (Theorem 8.3.2). As indicated at the beginning of this section, we shall not discuss the noncommutative version of the Gel'fand-Naimark Theorem in any detail, but shall only give a precise statement of it. The reader who is interested in a more detailed treatment is referred to [HoI' pp. 83-88; N, pp. 309-314; Ri, pp. 239-244]. More detailed discussions of B*-algebras are also available in these references.

10. B*-Algebras

280

Theorem 10.2.2 (Noncommutative Gel'fand-Naimark Theorem). Let A be a B*-algebra. Then there exists a Hilbert space V over ~ and a norm-closed symmetric subalgebra B of L(V) such that A is isometrically *-isomorphic to B. The invOlution in L(V) is naturally that given by the Hilbert space adjoint. Thus the theorem says that every B*-algebra is isometrically *-isomorphic to a C*-algebra. At a number of key points in the preceding chapters we have utilized Plancherel's Theorem (Theorem 7.2.1). This was the case, for example, in proving the regularity of Ll(G) (Corollary 7.2.3), in proving that LI(G) is a Tauberian algebra (Theorem 8.1.2), in proving that Ll(G) satisfies Ditkin's condition (Theorem 8.6.1), and generally throughout our discussion of spectral synthesis in LI(G). We are now in a position to prove this important theorem, and the next two sections are devoted to this task. Section 10.3 will be rather technical in nature, but it will provide us with the necessary machinery to establish P1ancherel's Theorem in the subsequent section. 10.3. The B*-Algebra ~(G). In this section we wish to discuss a number of detailed results about a certain commutative algebra of continuous linear transformations on the Hilbert space L2 (G), G being a locally compact Abelian topological group. We recall that L 2 (G) is indeed a Hilbert space with the inner product defined as {f,g) = JG f(t)g(t) dA(t) This algebra will be instrumental in proving Plancherel's Theorem in the next section. We begin with a definition. Definition 10.3.1. Let G be a locally compact Abelian topological group. If f (L1(G), then Tf (L(L 2 (G)) is defined by Tf(g) = f * g, g ( L2 (G).

10.3. The B*-Algebra A (G)

281

o

It is apparent from Example 1.2.8 that Tf E L(L 2 (G)) and that IITfll < lifll l , f E LI(G), since II Tf (g)1I 2 < II f ll 111g11 2, g E L2 (G). This definition of Tf in the algebra L(L 2(G)).

allows us to embed the algebra

L1 (G)

Lemma 10.3.1. Let G be a locally compact Abelian topological group. If ~: LI (G) - L(L 2 (G)) is defined by ~(f) = Tf , f E Ll(G), then ~ is a norm-decreasing *-isomorphism of LI(G) into L(L 2 (G)). Proof. It is easily verified that ~ is a norm-decreasing homomorphism of Ll (G) into L(L 2 (G)). To see that ~ is injective we suppose that f (LI(G) and Tf = O. If KeG is any compact set, then XK E L2 (G) n Lm(G), where XK denotes the characteristic function of K, and 0 = Tf(XK) = f * XK. Since by Proposition 4.7.2(i) f * XK is continuous, we hav~ in particular,

Hence from the regularity of the measure ~ E M(G), defined by d~(s) = f(-s) dA(s), we conclude that f = O. Thus ~ is an isomorphism. Finally, to see that ~ is an *-isomorphism it suffices to show that (T f )* = Tf *, f € L1(G), where f*(t) = f(-t). But, given f E Ll(G), if g,h E L2 (G), then {(Tf)*(g),h)

= (g,Tf(h)} = iG

g(t)f * h(t) dA(t)

= iG g(t)[iG f(t - s)h(s) dA(s)] dA(t) = iG h(s)[JG f*(s - t)g(t) dA(t)] dA(s)

= (f*

* g.h)

10. B*-Algebras

282

By a consequence of the Hahn-Banach Theorem [L, p. 89] and the Riesz Representation Theorem for Hilbert spaces [L, p. 390] we conclude that

(Tf )*

= Tf *·

0

Thus we see that ~[LI{G)] is a symmetric commutative subalgebra of L{L2 {G» that is *-isomorphic to LI{G). We now define the algebra Ao{G). Definition 10.3.2. Let G be a locally compact Abelian topological group. Then Ao(G) is the norm closure in L{L 2 (G» of ~[LI{G)] = {Tf I f E Ll{G»). Lemma 10.3.2.

group.

Then A CG) o

Let

G be a locally compact Abelian topological

is a commutative B*-algebra.

Proof. Clearly Ao{G) is a norm-closed commutative subalgebra of L(L 2 (G». It is symmetric, and so a B*-algebra, since ~[LI(G)] is symmetric and the involution in L{L 2 {G» is continuous. 0 From Corollary 10.2.1 we see at once that A (G) is a Tauberian, o self-adjoint, semisimple, regular, commutative Banach algebra. Next we wish to identify the maximal ideal space of A (G). o Lemma 10.3.3. Let G be a locally compact Abelian topological • group. Then 6{Ao(G» is homeomorphic to G. Proof. If w E 6{AO(G», then define Tw(f) = w{Tf ) = (w 0 ~) (f), f E L1{G). Evidently Tw E 6(L I (G», and so there exists a unique Yw E G such that

are such that wl(T f ) = w2 (Tf ), Moreover, if wl ,w2 E 6(Ao(G» f £ LI(G), then, since ~[LI(G)] is norm dense in Ao(G),. we conclude that wI = w2 . Thus the mapping from 6(Ao(G» to G, defined by w - Y , w E 6(A (G», is injective. Since the topologies on W • 0 6(A (G» and G are just the relative weak* topologies, it is o

10.3. The B*-Algebra Ao(G)

283

apparent that this mapping is also continuous. To see that the map• there exists ping is surjective we must show that, given y E G, some w E A(A (G)) such that y = y. o w With this in mind let w E A(A (G)) be fi~ed and suppose y E G. We need to define weT) for each TEA (G). If TEA (G) o 0 and (f) c: LI(G) is a sequence such that lim IITf - Til = 0, then n n n define gn = (.,y )e·,-y)f, n = 1,2,3, •••• Clearly (g 1 c: L1(G), Wo n n and . 0 0

= IG

w (T ) o gn

g Ct)(t,y ) dA(t) n Wo

= SG fn(t)(t,y) Fur~hermore,

d~Ct)

(n

= 1,2,3, ••• ).

some straightforward computation reveals that for each

m and n

sup liT lIhll2=1

&m

(h) - T (h) 112 gn

lIg

=

* h - g * hll2 IIh1l2=1 m n

=

sup IIr(·,y )('ry)f 1 * h - [(·'YWo)(·,-Y)fn1 * hll2 IIh1l2=1 Wo m

=

=

sup

sup IIf

llh1l2=1

m

* [(',-Y )(·,Y)h] - f Wo

*

[(',-Y

n

Wo

)(.,y)h1l 2

sup IIf * h - f * hll2 IIh1l2=1 m n

= IITf

- Tf II· m n Hence {T } is a Cauchy sequence in A (G), and so converges. In gn 0 particular, the sequence of numbers {w (T ») is convergent, and o gn we define weT) = lim w (T ). The preceding argument shows that n 0 gn weT) is well defined. Indeed, if (f~] C LlCG) is a second sequence such that lim IITf' - Til = 0 and g' = C",y )(·,-y)f', n = 1,2,3, •.• , n n n Wo n then, as above, we deduce that liT - T ,II = IITf - Tf ,II for each n, gn

gn

n

n

10. B*-Algebras

284

whence we have lim w (T ) = lim w (T ,). In this way, given • n 0 gn n 0 gn y E G, we obtain a well-defined linear functional w on A CG). o We claim, moreover, that w is a homomorphism on the following grounds: Suppose T,S ( A (G) and (f J and (f'l are sequences o

in Ll(G) such that Then the estimates

lim IITf n

n

~ IITS ... Tf

n - Til = 0

n

511

+

liT f

n

and

5-

n

lim IITf' n

n

511 = o.

Tf Tf' II

n n

< IIsliliT - Tf II + IITf 11115 - Tf ,1I n n n reveal that limn"TS - Tfn * f. 1l = O. Furthermore, if gn and g'n are defined as before, then itnis readily verified that g * g' = n n (·,YWb)C·,-y)(fn * f~), whence

weTS)

= lim

w (T ,) 0 gn * gn

= lim

w (T

= lim

w (T ) lim w (T ,) 0 gn n 0 gn

n

n

n

T ,)

gn gn

0

= w(T)w(S), as w E a(A (G»). Consequently w E A(A (G». Moreover, if 0 0 0 f E LI(G) and ~e take fn = f, n = 1,2,3, ... , then we see, on the one hand, that

= lim n

w (T 0

= lim IG

gn

)

fnCt) (t,y) dACt)

n

=

while, on the other hand,

IG

f(t)Ct,y) dArt),

10.3. The B*-Algebra Ao(G) Therefore y jective.

= y, W

285

and the mapping from

(G)) 0

~(A

to

..

G is sur-

An easy argument, which we shall leave to the reader, shows that the inverse mapping is continuous, and so ~(A (G)) is homeoo morphic to G. o

.

We now combine the preceding lemmas with the Commutative Gel'fandNaimark Theorem (Theorem 10.2.1) to obtain the following theorem~ Theorem 10.3.1. Let G be a locally compact Abelian topologica1 group. Then the Gel'fand transformation on A (G) is an iso.. 0 metric *-isomorphism of Ao (G) onto Co(G). Moreover, if f € Ll(G), then the Gel'fand transform of Tf € Ao (G) coincides \'1ith the Fourier transform of f. Proof. The only point that is perhaps not transparently clear is the last assertion. But, if f € Ll(G), then as seen in the proof of Lemma 10.3.3, we have

..

(T f ) (w) =

=

w(T f )

IG

..

f(t) (t,yw) dA(t)

= fey w)

(wEIl(A (G)), o

from which the assertion is evident.

c

Sinee Ao(G) is semisimple, the last portion of the theorem also provides a new proof of the semisimplicity of L1(G). Next we wish to use the B*-algebra Ao(G) to define a Fourier transform on a certain subspace Qf C(G) and then study some properties of this transform. Suppose TEAo (G). Then there exists a sequence (f 1 C L1(G) such that lim IITf - Til = O. Now in n n n some instances the sequence (f) may consist of continuous funen tions. For instance, if f € LI(G) and T = Tf , then there exists

10. B*-Algebras

286

a sequence (fn J c Cc (G) such that limn IIfn - fill = O. Hence. since z. is norm decreasing, we see that limnl/Tfn - Tfll = O. We wish to pay special attention to this situation in the case where we require that the sequence {f) converges uniformly on G. This n would occur. for example. if f E C (G). To be more precise. we c make the following definition: Definition 10.3.3. logical group.

Let

Then CAoCG)

G be a locally compact Abelian topois the set of all those

which there exist some T E Ao(G) with the following properties: (i)

lim IITf n

(ii)

n

- Til =

limn IIfn - fll co

and a sequence

f E CCG)

{fn}

C

Ll(G)

for

n C(G)

o.

= o.

Clearly CAo(G) ~~. Indeed, if f E Cc(G) and T = Tf , then with f = f, n = 1.2,3, •.• , we see that lim IITf - Til = 0 and n n n limn IIfn - fll co = O. Thus Cc (G) C CA0 (G). Moreover, we claim that f E CAo(G) is uniquely determined by the T corresponding to it. To see this, suppose that f,g E CA (G) and there exist some TEA eG) o 0 and sequences {f} and {g) in Ll(G) n C(G) such that n n limllTf - Til n n

= limllT

limllfn - fll m

= limllgn

n

gn

- TI/

=0

- gil co

= o.

and

n

If

f

~

g,

then obviously we have

n

h = f - g

limll(fn - gn ) - hll m

~

0 and

= o.

n

Since Cc(G) is norm dense in LI(G) and the dual space of LI(G) can be identified with Lm(G) [OSI' pp. 289 and 290], it follows easily that there exists some k E C (G) such that h * k ~ O. c Moreover, the estimate

10.3. The B*-Algebra Ao(G)

287

lI(fn - gn) * k - h * kll CD< lI(fn - gn)

-

hllCDIIklll

reveals that limn lI(fn - gn ) * k - h * kll CD = O. Thus, in particular, no subsequence of {(fn - gn ) * kl converges almost everywhere to zero, as h * k ~ O. However, since k (CcCG) c L2 (G) , (fnl C Ll(G), (g 1 C Ll(G). and lim IITf - T II = 0, we see from the estimate n n n h IICfn - gn) * kll2

= IITf

n

Ck) - Tg (k)1I 2

< IITf n

n

T

gn

II Ilk II 2

that limn IICfn - gn ) * kll2 = O. Standard results of integration theory [DS l , pp. 122 and 150] allow us to deduce that the sequence {(fn - gn) * k] has a subsequence that converges almost everywhere to zero, contradicting the preceding observation. Consequently f = g. Thus we see that f (Cho(G) is uniquely determined by the T E Ao(G) corresponding to f. To emphasize this fact and to indicate the correspondence between f E CA CG) and the appropriate o T (Ao(G) we shall frequently write f = fT' f E CAo(G). A similar argument employing the norm density of CcCG) in L2 (G) shows that. if f E CAo(G) and f = fT = fS' then T = S. The details are left to the reader.

.

It is now an easy matter to define a Fourier transformation on CA (G). Since f has already been used in a different context, we o shall use , to denote this transform. Definition 10.3.4. Let G be a locally compact Abelian topOq. • logical group. If f = fT E CAoCG) , then we set t = T, where T denotes the Gel'fand transform of TEAo (G). Clearly 'E Co (G) Lemma 10.3.4.

group. Ci)

by Theorem 10.3.1.

Let G be a locally compact Abelian topological

Then CAoCG)

is a translation-invariant linear subspace of C(G).

10. B*-Algebras

288

(ii)

Cc (G)

C

Ll(G) n C(G)

CA0 (G).

C

(iii) If f = fT (CAo(G) and f*(t) = f(-t), t E G, f* ( CA (G) and f* corresponds to T* E A (G). o 0 (iv)

If f = fT E CAo(G) ,

(v) If f E CAo(G) the Fourier transform of

then

n Ll(G),

q.

then

= f,

t

where

(b)

o

f

•

= f,

IItii

Proof.

GO

•

f

denotes

f. defined by f - t, • CAo(G) to Co (G)

(vi) The aapping from CAo(G) to Co (e) , f E CAo(G) , is the unique iinear mapping from such that (a)

then

f E CAo(G)

n Ll(G).

= IITII, f = fT ( CA0 (G).

The proofs of parts (i) and (ii) are left to the reader.

If f = fT E CAo(G) , then there exists a sequence {fn ) C Ll(G) n C(G) such that limn IITfn - Til = 0 and limn IIfn - fll = O. From Lemma 10.3.1 we see that Tf * = (T f )*, and so, by Proposition 10.1.I(vi), n n CD

limllTf* - T*II n

n

= liml/(Tf n

= 1imllTf n

)* - T*II n

-

Til

n

=0 and limllf*n - f*1I = limllfn - fll = o. n n Thus f* E CAo(G) and f* = f *. Part (iv) is immediate from Theorem 10.3.1. CD

CD

T

To prove part (v) suppose f (CAo(G) n L1 (G). Then, on setting f = f, n = 1,2,3, .•• , we see at once that lim IITf - Tfll = 0 and n n n limn IIfn - fll = 0, whence we conclude that f = fT. Hence, by f CD

10.3. The B*-Algebra

Ao(G)

Theorem 10.3.1, we have

289

t = (Tf )- = f. o

Finally, it is now apparent that the mapping f - f, f £ ChOCG) , is linear and satisfies conditions (a) and (b) of part (vi). Suppose f -1, f E CA (G) is another such mapping. If f = fT £ CA (G) o 0 and (f) C Ll(G) n C(G) is such that lim IITf - Til = 0 and n n n limn IIfn - fll CD = 0, then from parts (ii), (iv), and (v) we see that

111' - til CD-< 111 - 1nCD II + IIrn - t nCD II + II!n - '"CD =

!Ir - Tf II n

from which it follows that 1 lemma.

+

= f.

liT f

n

- Til

(n = 1,2,3, ••• ),

This completes the proof of the

o

In particular, the last part of the lemma says that the definio tion of f, f E CA (G), is the only way we can extend the Fourier o transform on CA (G) n Ll(G) to all of CA (G). In the next seco 0 tion we shall use this Fourier transform on a subspace of CAo(G) (1 L2 (G) to define an appropriate Fourier transform on all of L2 (G) and thereby obtain Plancherel's Theorem. Before this, however, we need some rather technical results about CA (G). o

Lemma 10.3.5. Let G be a locally compact Abelian topological group. If f E CA (G) and , is real valued, then f(O) is real. a

Proof. Suppose f = fT. Then of = T, and, since the Gel'!and transformation on0 A0 (G) is an *-isomorphism, _ we _see that (T*) = W Q q T = f =~, as f is real valued. Thus (T*) = T, and so T = T* since A (G) is semisimple. If (f 1 C LlCG) n C(G) is such that o n limn IITfn - Til = 0 and limn IIfn - fll = 0, then we also have limn IIT£*n - Til = limn IITf*D - T*II = O. Let gn = Cfn + f n*)/2, where, as usual, f n*(t) = f n (-t), t E G. Then (g] n C Ll(G) n C(G), and it is evident that limn liTgn - Til = 0 and limnn IIg - (f + f*)/211 = o. However, f = fT is uniquely determined by T, and so we conclude that f = (f + f*)/2. Thus frO) = [frO) + f(0)]/2 is real. o CD

CD

10. B·-Algebras

290

Lemma 10.3.S is also valid if we replace real by nonnegative throughout its statement. Lemma 10.3.6. Let G be a locally compact Abelian topological o • group. If f E CAo(G) and fey) > 0, Y E G, then f(O) > o. Proof. Let f = fr. From Lemma 10.3.S we know that f(O) is real. Suppose f(O) < O. Then, since f is continuous, there exists a symmetric open neighborhood U of 0 that has co,pact closure for which If(t) - f(O)1 < If(0)1/2, t £ U. Recall that U symmetric means that U = -U. Let g E Ll(G) n L2 (G) be such that (a)

g * g* E Cc(G).

(b)

The support of g * g*

(c)

IG g * g*(s)

dA(s)

is contained in U.

= 1.

Such an element g can be obtained in the following manner: Let W be an open symmetric neighborhood of the identity in G such that W+ WC U and set gl = Xw' the characteristic function of W. Then g = gl/[iG gl * gies) dl(s)]1/2 has the indicated properties. We claim that f * g * g* £ CAo (G). Indeed, from Proposition 4.7.2(i) we see that f * g * g* E C(G). Furthermore, let (fn) C L1 (G) n C(G) be such that limn 1I1fn - 111 = 0 and limn IIfn - fll CD = 0, and set hn = fn * g * g*, n = 1,2,3, ..• , and S = TTg * g*. Obviously (hnl C LI(G) n C(G), while the estimates - S

II

=

liTf

n

* g * g*

- TTg

< 1I1f - TlIlITg * g*1I n

and

* g* II (n =

1,2,3, ••• )

10.3. The B*-Algebra A (G)

291

o

show that f * g * g* E CA (G) and corresponds to S = TT * o g * g in Ao(G). Moreover, from Lemma lO.3.4(ii) and (v), we see that o

(f * g * g*) (y)

= (TTg

.

•

* g*) (y)

= T(y)(g

-

* g*) (y)

0·2

= f(y)lg(y)1

>0

(y E G).

Thus, in particular, from Lemma 10.3.5 we see that is real. Thus, on the one hand, the estimate If * g * g*(O) - f(O)1

= IIG

(f * g * g*)(O)

f(s)g * g*(-s) d1(s) - f(O)1

< Iu If(s) - f(O)lg * g*(-s) d1(s) < If(O)l 2

reveals that real.

f * g * g*(O)

~

f(0)/2 < 0,

q.

as

f * g * g*(O)

is

• • •

But, on the other hand, try) = T(y) > 0, y E G, and TEe (G) entail that (T)I/2 is a well-defined real-valued function in °Co(G). Since by Theorem 10.3.1 the Gel'fand transformation on Ao(G) is surjective, we deduce that there exists some PEA (G) such that • = (T) • 1/2 . Moreover, since A (G) is semisimple, 0 (T) • 1/2 P is real o valued, and the Gel'fand transformation is an *-isomorphism, we see that P is self-adjoint and p2 = T. Now let (gn) C Ll(G) be such that lim liT - pil = O. Since P is self-adjoint, we may assume, n gn without loss of generality, that g* = g , n = 1,2,3, ... , as was n n done in the proof of Lemma 10.3.5. Then for each n we have

292

10. B*-Algebras

lim liT - T *11 = O. In partin gn * gn cular. we must also have lim IIT£ - T *11 = O. Finally then, n n gn * gn since g * g* (; Cc (G), g E L2 (G), and limn IIfn - fll m = 0, we see that whence we conclude at once that

f * g * g* (0)

= iG

g * g*(-s)f(s) dA(s)

= lim iG

g * g*(-s)fn(s) dA(s)

= lim fG

Tf

n = lim f n * g * g*(O) n = lim Tf (g) * g*(O) n n n

(g)(s)g(s) dA(s) n

= lim

{Tf (g),g)

= lim

{Tg

= lim n

g * g* * g * g*(O) n n

= lim

(g

n n

n * g*(g),g} n n

n

= lim fG

* g) * (g

n

(g

n

= lim

IG

= lim

(lign

n

n

* g)*(O)

n * g)(s)(gn * g)*(-s) dA(s)

(gn * g)(s)(gn * g)(s) dA(s)

= lim rliTg n

n

n

*

g1l2)2

(g)1I 212

= rll p (g)1I 212 > O.

This, however, contradicts the previous observation that f

Therefore

f(O) >

o.

* g * g*(O) < o. [j

10.3. The B*-Algebra A (G)

293

o

We shall have need of one more result about CA (G). But in o order to prove it we need to introduce a new topology on G and establish a preliminary result. Lemma 10.3.7. Let G be a locally compact Abelian topological • group. For each ~ > 0 and each compact set KeG define N(K,~)

= {t

Then the family of sets

•

1 t E G, 1Ct,y) - 11 <~, y (K). (N(K,~)J,

where

~

> 0 is arbitrary and

KeG is an arbitrary compact set, forms a neighborhood base at the identity of G for a topology that is weaker than the given topology on G. We shall see subsequently that the topology obtained as in the lemma actually coincides with the given topology on G. The proof of the lemma is left to the reader. The argument is essentially the same mutatis mutandis as the one given in the first half of the proof of Theorem 4.7.5, the crucial point being that the mapping • • from G X G ~ r, defined by (t,y) ~ (t,y), t E G and y EGis continuous. Lemma 10.3.8. Let G be a locally compact Abelian topological group. If KeG• is compact, 6 > 0, and a > 0, then there exists some g E C (G) such that c (i)

get) > 0, t E G.

(ii)

• • > 0, y E G. g(y)

(iii)

•

Ig(y) - al < 6, y E K.

Proof. From Lemma 10.3.7 we see that N(K,6/a) is an open neighborhood of the identity in G. Let U c N(K,6/a) be an open neighborhood of the identity in G that has compact closure and let W be a symmetric open neighborhood of the identity such that W + We U. Let g = c(Xw * Xw), where c is so chosen that

10. B*-Algebras

294

IG

g(s) dA(s) = a.

Since Xw * Xw is nonnegative, we see that c > O. Evidently g E Cc(G) , the support of g is contained in W+ W, get) ~ 0, t E G, and

•

•

g(v) = c(Xw * Xw) (V) =

c(Xw *

XW) • (V)

• 2 = cIXw(v)1

•

(V E G).

>0

Finally, if y E K,

•

19(y) - al

then

= IIG

g(s)(s,V) dA(S) -

~Iu g(s)l(s,v) ~

sup

s E N{K,6/a)

< 6.

11

I(s,v) -

IG

g{s) d1{s)1

dA(S)

11 IG

g{s) dA(S)

o

couple of additional remarks about this lemma are in order. Clearly the function g is of the form g = f * f*, where • = I-fey) 12 > 0, V E G. • Moreover, f e Ll(G) n L2 {G) n L.(G) and g(V) it is not difficult to see that we can even demand that f ( C (G). c Indeed, in the proof of the lemma we need only replace Xw by Xwl * XWI ' where WI is a symmetric open neighborhood of the identity such that WI + WI C W. We shall have need of these observations in the next lemma and in the next section. A

Lemma 10.3.9. Let G be a locally compact Abelian topological group. If ho E Cc (G) is real valued, then for each e > 0 there exist h,k E CA (G) such that

-

o

(i)

~,~ E C (G). c

10.3. The B·-Algebra Ao (G) (ii) (iii) (iv)

o

295

0

hand k are real valued.

o '"'"----~

0

..

key) < h (y) < hey), y € G. -

0

-

0
Proof. Let the compact support of h be K and let a = 2 o and 6 = 1/2. By bemma 10.3.8 there exists some gEe (G) such o • .... 0 c that g (t) > 0, t e G, g (y) > 0, y E G, and Ig (y) - 21 < 1/2, o 0.. 0 y E K. In particular, since g is real valued, we have .. 0 go(y) > 3/2 > 1, y E K. Applying Lemma 10.3.8 again, this time with a = I and 6 > 0 arbitrary, we obtain a function gEe (G) • • • c such that get) > 0, t ~ G, g(y) ~ 0, y € G, and 19(y) - 11 < 6, Y E K. In particular, 1 - 6 < g(y) < 1 + 6, y E K. We shall specify 6 more precisely in a moment.

..

..

First, however, since ho E Cc (G) and the Gel'fand transformation on A (G) is a surjective isomorphism by Theorem 10.3.1, we o .. deduce the existence of a unique TEA (G) such that T = h. Let o

0

Tl = TTg+ 6g and T2 = TTg-~go ~ . We claim that Tl and T2 determine elemen~s of CA (G) for any 6. To see this it clearly suffices o to show that TT and TT determine elements of CA (G). But g go 0 suppose {f) c:: Ll(G) is such that lim IITf - Til = o. As observed n n n after Lemma 10.3.8, we may assume, without loss of generality, that g = f • f*, f E Ll(G) n L2 (G) n Lm(G). Consequently (fn • g) belongs to Ll(G) n C(G) and, using HHlder's inequality, we find from the estimates IIfn * g - f m • gilm = lI(fn - f m) • f * f*1I m < lI(fn - fm) • f1l211f*1I2 < IITf - Tf n

II

m

Uf1l211f*1I2

(n,m

= 1,2,3, ... )

and

IITf * g n

11 II < g -

IITf - TIIIITg II n

(n =

1,2, 3, ... )

10. B*-Algebras

296

that lim IITf - 17 II = 0 and that (f * g) cenverges uniformn n * g g n Iy to some function in C(G). Hence TTg determines an element of CAo(G). A similar argument shows that the same is true of TT . go Thus TT 6 and TT 6 determine elements of CA (G), g+ go g- go 9 o. • which we denote by hand k, respectively. Clearly h = T(T +6 ) •• 9 • • 0 g go = h (g + 6g) and k = h (g - 6g), whence we see that hand 000 00. k are real-valued functions in C (G). Moreover, if y € K, then • • c since g (y) > I and g(y) + 6 > 1, we have o

o

hey)

••

= ho (y)[g(y)

+ 6g 0 (y)]

o

0

whereas if y ~ K, then h(y) = 0 = h (y). Thus h(y) > h (y), • 0 .0 0 y (G. Similarly k(y) < h (y), y E G. Hence parts (i) through (iii) 0 of the lemma are established for any 6 > O. However, to obtain part (iv) we must put some additional restrictions on 6. Suppose go inequality,

= f 0* 0 f*, 0 f

E Ll(G)

n Lm (G). Then, by HHlder's

Ifn * go(O)l = Ifn * f 0 * f~(O)1 = ITf (fo> * f*(O)1 0

n = lIG Tf (fo)(s)£o(s) dA(s)1 n (n = 1,2,3, ••• ). < IITf (fo)1I2l\ f oll2 n Since (I\Tf III is a bounded sequence because lim UTf - Til = 0, n n n we conclude that {f * g (0») is a bounded sequence. Set n 0 x = supn Ifn * g0 (0)1 and suppose 0 < 6 < e/6x. Now from the proof that TT 6 and TT 6 determine elements of CAoCG) it is g+ go g- go immediately apparent that

10.4. Plancherel's Theorem

297

limllfn * (g n

+

6g 0 ) - hI! CD

=0

and limllfn * (g - 6g 0 ) - kll n

Q)

Thus there exists some positive integer n

=

o

o. such that, if n>n, -

0

then IIfn * (g

+

6g 0 ) - hll

Q)

< £3

and

JJ f n Therefore, for

Ih(O)

n

>n , -

0

* (g - 6g0 ) - k II CD < £3· we have

- k{O)1 < Ih(O) - f n * {g + 6g o )(0)1 + Ifn * (g + 6g 0 ) CO) - f n * (g - 6g 0 ) (0) I +

Ifn

* (g - 6g0 ) (0) - k(O)1

2£ <-+ 261f * go (0) I 3 n 2£ - 3

<-+ 26H.

< However, since that

£.

0 0 .

hey) - key) > 0, y E G,

h(O) - k(O)

~

0,

whence

we see from Lemma 10.3.6

0
c

10.4.

Plancherel's Theorem. In this section we shall apply the results developed in the preceding section to prove Plancherel's Theorem. The first step in this direction is an inversion theorem that is of considerable interest in its own right since it allows

us to recover certain

f E CA (G) o

from a knowledge of

t.

10. B*-Algebras

298

Theorem 10.4.1 (Inversion Theorem). Let G be a locally compact Abelian topological group and let A be a given Haar measure • can be so chosen that, if on G. Then a Haar measure ~ on G f E CAo(G) and t E Cc(G), then f(t) =

fa

ley) (t,y) d~(v)

(t E G).

Proof. The crux of the proof is to construct an appropriate • translation invariant nonzero positive linear functional on Cc(G) that will yield the Haar measure we require. By such a functional • to C that is F we mean, of course, a linear mapping from C (G) c • not identically zero and such that F(f) > 0 whenever f £ C (G) • c. and f(V).> 0, V E G, and such that F[TV(f~] = F(f), f E Cc(G) and V E G. With this in mind, let g E C (G) be real valued. c Then from Lemma 10.3.9 we see that sup{k(O) , k E CA (G), ~(V) < g(y), V E o -

= inf{h(O)

Gl

, h E CAo(G), ~(V) > g(V), V E Gl.

We denote this common vAlue by F(g). In this way we obtain a map• -~ where CR(G) • ping F: CR(G) denotes the space of real-valued c • c continuous functions on G with compact support. It is easily verified from the definition of F that F is linear and F(g) > 0 • Moreover, if f E- CA (G) whenever g E CR· (G) and g(V) > 0, V E G. oR. c 0 and f E Cc(G) , then, since in this instance we can take h = k = f in Lemma 10.3.9, we see that F(!) = f(O). We can extend F to • in the obvious manner: if gEe (G), • all of Cc (G) then c g = Re(g) + iIm(g), and we set F(g) = F[Re(g)] + iF[Im(g)]. Hence F is a positive linear functional on C (G) such that FC!) = f(O) c whenever f E CA (G) and tEe (G). Furthermore, we claim that o c F is not identically zero. Indeed, by Theorem 10.3.1, there exists some T E AO(G) , T ~ 0, • • such that TEe (G). Let gEe (G) C L2 (G) be such that c c IIT(g)1I2 > 0 and set f = T(g) * T(g)*, where, as usual, T(g)*(t) =

10.4. Plancherel's Theorem

299

From Proposition 4.7.2(ii) we see that fEe (G). Actuo ally, f E CA (G). To see this, let {f) c L1 (G) be such that o n lim IITf - Til = O. Moreover, lim IIT£* - T*II = 0 by Lemma 10.3.1. n n n n Now (fn * f*n * g * g*) c L1 (G) n erG) and for n = 1,2,3, ...

T(g)(-t).

IIfn * f*n * g * g* - T(g) * T(g)*11 m =

lI(fn * g) * (fn * g)* - T(g) * T(g)*11 m

=

IITf (g) * Tf (g)* - T(g) * T(g)*lI co n

n

< IITf (g) * Tf (g)* - Tf (g) * T(g)*lI co n

+

n

n

IITf (g) * T(g)* - T{g) * T{g)*lI co n

:s. IITf (g)* - T{g)*1I 2I1 Tf (g)1I 2

n n + IIT(g)*1I 2i1 Tf (g) - T(g)1I 2 n

= IITf

n

(g) - T(g)1I 2 [IITf (g)1I 2

+

IIT(g)1\2]'

n

Hence we conclude that limllf * f* * g * g* - T(g) * T(g)*11 n n n m Furthermore, for

= O.

n = 1,2,3, ...

IITfn * f*n * g * g*

whence we have limllT f * f* * g * g* - TT*Tg * g*1I = 0 n n n by the continuity of multiplication in a Banach algebra. Consequently f = T(g) * T(g)* ~ CA (G), and f corresponds to TT*T * E A (G). q • ..0. • g * g 0 However, f = T(T*) g(g*) E Cc (G), and so F(t)

= f(O) = T(g)

* T(g)*(O)

10. B*-Algebras

300

= fG T(g) (s)T(g)(s) dA(s)

= [IIT(g)1I 2]2 > O. Hence F is not identically zero. Finally, we claim that F is translation invariant. Indeed, by the same sort of argument as we used several times previously, it is not difficult to show that, if k (CA (G), then (·,w)k ( CA (G), • 0 0 0 0 w E G, and [(·,w)k] = T (k). The details are left to the reader. -w • Thus, since k(O) = [(.,-w)k](O), w £ G, we deduce that, given • • g (C R(G), we have for each w € G c F[Tw(g)]

G)

= sup{k(O)

IkE CA (G), ~(y) < T (g)(y), Y E o - w

= sup{k (0)

g(y), y E G} I k (CAo (G), T-w (k)(y) < -

o

•

o

•

= sup{k (0) I k (CAo (G), [(.,-w)k] (y) <- g(y), y E G) = sup{k (0)

I

o

•

k ( CA (G), k (y) < g (y), y E G) o -

= F(g). It is now apparent that

F is translation invariant.

An appeal to the Riesz Representation Theorem for positive linear functionals [HR 1 , p. 129] shows that there exists a unique posi• tive regular Borel measure ~, that is, a Haar measur~on G such that F(g) = Consequently, if f E CA (G) o f(O) = However, if t E G and

fa g(y) and

o

(g E C (G)).

c

•

f E C (G), c

fa t(y)

f ( CA (G), o

•

d~(y)

then

d~(y). then an easy argument reveals

10.4. Plancherel's Theorem

301 0 0 .

that T_tCf) E CAo(G) and that [T_tCf)] (y) = (t,y)f(y), y ( G, o • from which we deduce that, if f E CAoCG) and f E CcCG), then f(t)

= T_t(f) (0) = fa = fa

o

[T_t(f)] (y) d~(y) o

(t E G).

f(y) (t,y) dftCy)

This completes the proof of the inversion theorem.

o

The following corollary is now immediate since, by Lemma 10.3.4 (v) ,

o

•

f = f

f E CAo(G) n Ll(G).

whenever

Corollary 10.4.1.

Let

G be a locally compact Abelian topolo-

gical group and let A be a given Haar measure on G. Then a Haar • can be so chosen that, if f E CAo(G) n LI(G) measure ~ on G • • and f E C (G), then c fCt) =

fa

•f(y) (t,y)

d~(y)

(t ( G).

Note that in the statement of the corollary we are tacitly assuming that f is a continuous function; that is, we are considering f as an element of C(G). If f is thought of as an element of LICG), that is, as an equivalence class, then the formula in the corollary holds only almost everywhere. Utilizing this inversion theorem, we can now prove Plancherel's Theorem. Theorem 10.4.2 (Plancherel's Theorem). Let G be a locally compact Abelian topological group and let A be a given Haar measure on G. If

v = (f I then a Haar measure

(i)

f E CA (G) o ~

on

n L2 (G) , , ( Cc (G)],

• can be so chosen that G

V is a norm-dense linear subspace of

L2 (G).

10. B·-Algebras

302 (ii) (iii)

o

V = (f0 , f E V] IIfll2

= 11'11 2,

• is a norm-dense linear subspace of L2 (G).

f E V.

(iv) The mapping f ~ I, f E V, from V to ~ can be uniquely • extended to a linear isometry of L2{G) onto L2{G). o

Proof. Obviously V and V are linear subspaces of L2 (G) • respectively. To see that V is norm dense in L {G) and L2{G), 2 let g E L2 (G) and I > O. Then there exists some f E Cc{G) such that IIf - gll2 < 1/2. Moreover, we ~laim that we may assume without loss of generality that f = fl * f2 * f3' fk E Cc(G), k = 1,2,3. Indeed, from the proof of Theorem 4.7.2, if (u 1 is a family of symmetric open a neighborhoods of the identity in G that have compact closures and are such that n(tUa = {oj, then we know that {u] /A(U(t)1 a = {~. 'U a is an approximate identity for LICG). It is easily seen that, if f E Cc (G), then lima''If - f * u(tCII II = 0 and that (ua * ua1 c: Cc (G) is also an approximate identity. After combining these remarks it becomes apparent that we can assume f has the indicated form. We leave the details to the reader.

• and, by Theorem 10.3.1, the Gel'fand Now since •fl E Co(G) transformation on A (G) is a surjective isometry, we see that o •• there exists some TEAo (G) such that T E Cc (G) and

liT - fIll

=

liT - Tf II < 211f : f 1

CII

2

II .

3 2

Thus

c

<2 + IITf (f2 * f3) - T(f 2 * f 3)11 2 1

c

<-2

+

IITf - Tllllf2 * f3112 1

c c + - = c. <2 2

10.4. Plancherel's Theorem

303

However, T(f 2 * f3) E V. Clearly T(f 2 * f3) {h ) C Ll(G) is such that lim 11Th - Til = 0, n

n

n

lim 11Th f f - ITf f II = O. n n * 2 * 3 2 * 3

E L2 (G), and if then it follows that

But the estimates

IIhn * f2 * f3 - hm * f2 * f311CD< IIhn * f2 - hm * f211211f3112 =

11Th (f 2) - Th (f2)1I2 I1f 311 2

n

m

~ 11Th - Th IIl1f2112I1f3112' n

m valid for n,m = 1,2,3, ••• , reveal that (hn * f2 * f3 1 is a Cauchy sequence in C(G). Thus there exists some h E C(G) such that limnllhn * f2 * f3 - hUCD = O. However, we also have Inimllhn * f2 * f3 - T(f 2 * f 3)11 2 = limliTh (f2 * f3) - T(f 2 * f 3)11 2 n

n

= 0,

as limnUTh - Til = O. Consequently, by a result of integration theory [DSl~ pp. 122 and ISO], we deduce that (hn * f2 * f3 1 has a subsequence that converges almost everywhere to T(f 2 * f 3). Hence T(f 2 * f3) = h almost everywhere, and so we may assume that T(f 2 * f3) E L2 (G) is the continuous function h. Thus T(f 2 * f3) belongs to CAo(G) n L2 (G), as (hn * f2 * f3 1 C Ll(G) n C(G). Moreover, it is apparent from the preceding argument that T(f 2 * f3) o

--

corresponds to _TTf 2* f 3 E A0 (G), and so T(f 2 * f3) = TTf 2* 3 f belongs to Cc(G). Therefore T(f 2 * f3) ~ V and V is norm dense in L2 (G).

0 - -

To see that V is norm dense in L2 (G) we let g E L2 (G) and e > O. Since Cc(G) is dense in L2 (G) and the Gel'fand transfor_ _ we see that _there exists some mation on Ao (G) is surjective, TEAo (G) such that _ TEec (G) and lIg - TII2 < e/2. Let the compact support of T be K and apply Lemma 10.3.8 with a = 1 and 6 = e/[2l1TII the existence of some fEe c (G: _ CD1)(K)1/2] to _deduce 1/2 for which I fey) - Il < el [211TIl 1)(K) ], y E K. The Haar measure CD 1) on G can, at this point, be any such measure. As indicated in

10. B*-Algebras

304

the remarks following Lemma 10.3.8, we may assume that f = f o * f*, 0 where f o (C c (G). Repeating the same argument as before, mutatis 0 __ mutandis, we see that T(f) = TCfo * f*) E V and TCf) = Tf. Con0 sequently

liT - TfU2

=

[fa

IT(y)1 2 cll - fcy)12) dll(y)]1/2

=

[f K

IT(y)12(ll - f(y)12) dll(y)]1/2

r

~ II ll..ll(K)I/2[ 11.11 2 T

e

c 1/21 1}(K) co

= 2' whence

e

e

<-+2 2

= e. Thus

oV is norm dense in

L2 (G).

-

Next we choose Haar measure r. on G so that Theorem 10.4.1 is valid. If f £ V, then f . f* ( CA CG) and (f * f*)o = o 0 q~ _ 0 f(f·) = f f E C (0). Hence, by the Inversion Theorem, we see that c

= fG

=f = fa =

f(s)£*(-s) dA(s) * f*(O) (f

*

f*)

fa !(y)JCY)

o

(y) d1}(y) dll(y)

= (11," 2)2, from which it follows at once that the mapping

o

f - f, f (V,

is a

10.4. Plancherel's Theorem

305 o

V onto V.

linear isometry from

Thus parts (i) through (iii) are proved, and the proof of part (iv) is now apparent.

o

It should be remarked that part (ii) of Plancherel's Theorem follows immediately from parts (i) and (iii). Nevertheless we gave the direct proof of part (ii) to emphasize the fact that the density o .. of V in L2 (G) is independent of the choice of Haar measure on G.

-

In accordance with the version of Plancherel's Theorem introduced in Section 7.2 we shall call the unique extension of the mapo ping f - f, f (V, to all of L2 CG) the Plancherel transformation and denote, for the moment, the Plancherel transform of f E L2 (G) o by f. The same proof as that given for Corollary 7.2.1 establishes Parseval's Formula. Corollary 10.4.2 (Parseval's Formula). Let G be a locally compact Abelian topological group and let A be a given Haar measure on G. If Haar measure ~ on G is so chosen that Plancherel's Theorem is valid, then

..

Several other useful properties of the Plancherel transformation are easily proved. We shall prove only the last one stated in the next corollary. The others are left to the reader. Corollary 10.4.3. Let G gical group, let A be a given a Haar measure on G so chosen If f E L2 (G), then for almost

..

_0

Ci) Cii)

be a Haar that all

locally compact Abelian topolomeasure on G, and let ~ be Plancherel's Theorem is valid .

..

y EG

0

= fC-y)

fCy) o

Cf*) Cy)

0

= fCy)

Ciii) [C·,w)f]oCY)

= tCY

- w), w E

G.

10. B*-Algebras

306

Moreover, if f,g E L2 (G), then (fg)-(y) = t * g(y), where y E and (fg) denotes the usual Fourier transform.

-

G

Proof. If f,g E L2 (G), then evidently fg E LI(G). Hence from Parseval's Formula and parts (i) and (iii) of this corollary we see that (fg) " (y) =

IG

f(t)g(t) (t,y) dAft)

=fG

f(t)iCtf(t,y) dAft)

fa =Sa

o

o

I

f(w)g(w - y) d~(w)

=

~(w)g(y - w) d~(w)

=f *

-

0

g(y)

(y E G).o

The present version of Plancherel's Theorem is not quite the same as the one introduced in Section 7.2 since V need not be a subset of Ll(G) n L2 (G). However, the next theorem combined with Theorem 10.4.2 immediately yields the previous form of Plancherel's Theorem (Theorem 7.2.1). Here "f will denote the usual Fourier transform. Theorem 10.4.3. Let G be a locally compact Abelian topological group, let A be a given Haar measure on G, and let \ be a Haar measure on G so chosen that Plancherelts Theorem is valid. If

-

then (i)

Vo

is a norm-dense linear subspace of

L2 (G).

(ii)

" Vo

is a norm-dense linear subspace of

" L2 (G).

Proof. We note first, by Lemma 10.3.4(v), that of = "f, f € Vo . To prove part (i) it suffices to show that V2 = (gh I g,h € VJ is

10.4. Plancherel's Theorem

307

norm dense in L2 CG). This is so because, if g,h E V, then o 0 .. I.h E CAoCG) n L2 (G) and g.h E Cc(G). whence. by Lemma 10.3.4(ii) and (v) and Corollary 10.4.3, we see that gb E CCG) n LICG) n L2 (G) o .. 2 C Cho(G) n L2 (G) and (gb) = (gb) = g * h E C (G). Thus V c V . o 0 2co . . 0 Moreover, since it is apparent that V * V = (V) C V and since o tbe Plancberel transformation is an isometry, we see that we can prove botb parts (i) and (ii) by showing tbat ~ * ~ is norm dense in L2(G).

0,..

.

.

So suppose f E L2 (G) and let proof of Theorem 10.4.2, there exist IIf - g * hll2 < e/3. Furthermore, by • dense in L2 (G) , and so there exist

..

e > O.

Then, as noted in the g,b E C (G) such that c 0 Plancberel's Theorem, V is 0 gl,h l E V such that

IIg - gl112 < e/ 3(U h ll l + 1) and lib - hll12 < '/3(lIg 1I1 l ing tbese estimates, we see tbat IIf - gl * hlll2 ~ IIf - g * hll2 +

<;

+

+ 1).

Combin-

IIg * h - gl * hil 2

IIg l * b - gl * hlll2 +

IIg - g111211hlll

+

1I1111111h - b l ll 2

C I I <-+-+-

333

=

I, 0 0 "

whence we conclude that V * V is norm dense in L2 (G). (i) and (ii) of the theorem are proved.

Thus parts

o

We restate the previous form of Plancherel's Theorem for the sake of easy comparison. Theorem 10.4.4 (Plancherel's Theorem). Let G be a locally compact Abelian topological group and let A be a given Haar measure • and a linear subon G. Then there exist a Haar measure ~ on G space Vo of L2 (G) such that

308

10. B*-Algebras

(ii)

V

is norm dense in

L2 (G) •

(iii)

V

•

is nonn dense in

L2 (G) .

(iv)

0

0

IIfll2

= II •f 1l 2 ,

•

f EV• 0

• • can be (v) The mapping f -. f, f E V , from V to V 0 0 0 • uniquely extended to a linear isometry of L2 (G) onto L2 (G). Naturally, the Haar measure ~ is so chosen that Theorem 10.4.2 is valid and Vo = V n Ll(G). Clearly the extension of the Fourier transformation given by this theorem coincides with the one given by Theorem 10.4.2. Actually it is possible to prove that f = t for any f E LI(G) n L2 (G) and that one can take Vo = LICG) n L2 (G) in the preceding theorem. We shall not, however, take the time to do this. As in Section 7.2, it is standard practice to denote the Plancherel transform of f by i, rather than t. For obvious reasons it was not desirable to do this for the greater part of this section, but we shall utilize this notation for the remainder of this volume. There are other proofs of Plancherel's Theorem that do not make use of B*-algebras or the Commutative Gel'fand-Naimark Theorem. These proofs generally utilize some sort of inversion theorem and Bochner'S Theorem on positive definite functions (see, for example, [N, pp. 409 -415; Ru l , pp. 17-27]). Other approaches are discussed, for instance, in [HR 2 , pp. 324-326; We, pp. 111-115]. We chose the present approach to Plancherel's Theorem since it is closer to the general theme of this volume than other methods. Other discussions of this approach, originated by J.H. Williamson, can be found in his original paper [WI] and in [Hy, pp. 148-163]. 10.5. The Pontryagin Duality Theorem. We have observed that, • if G is a locally compact Abelian topological group and G

10.5. The Pontryagin Duality Theorem

309

is the set of continuous homomorphisms of G into the unit circle r in the complex plane, then G can be provided with algebraic operations and such a topology that it becomes a locally compact

-

Abelian topological group, called the dual group of G. Moreover, we have given certain examples that suggest that the dual group of G, denoted by (G), is topologically isomorphic to G. The main purpose of this section is to prove this assertion, generally kno~~ as the Pontryagin Duality Theorem. The results referred to were discussed in Section 4.7.

-.

-

The first step in proving the indicated theorem is to observe

-

the existence of a natural isomorphism of G into (G). Indeed, given t (G, we define a(t)(y) = (t,y), y (G. It is easily verified that aCt) is a homomorphism of G into f. The continuity of aCt) follows from the fact that, given e > 0, the set U({t),e) = {y lyE G, l(t,y) - 11 <.) is an open neighborhood of the identity in G, by Theorem 4.7.5. The mapping a: G - (G) so defined is clearly a homomorphism. The injectivity of a follows at once from Corollary 4.7.3, while an elementary argument utilizing Theorem 4.7.5, Lemma 10.3.7, and the simple observation that for each e > 0 and each compact set KeG we have

-

.-

.-

-

N(K,e)

= (t =

I t E G, I(t,y) -

11

< s, y € K)

a-l[{~ I x ( (G)-, lx(y) - II < s, y ( K)]

reveals that a is continuous. In view of this fact and Lemma 10.3.7 it is apparent that to prove a is a homeomorphism we need only show that the sets N(K,s) in G form a neighborhood base at the identity for the topology of G. This strengthening of Lemma 10.3.7 is the second preliminary step to proving the Pontryagin Duality Theorem. Theorem 10.5.1. cal group. For each

N(K,s)

= {t

Let G be a locally compact Abelian topologis > 0 and each compact set KeG define

-

I

t

E C, I(t,y) -

11

<

S,

y E K).

310

10. B·-Algebras

..

Then the family of sets (N{K,e)}, where e > 0 is arbitrary and KeG is an arbitrary compact set, forms a neighborhood base at the identity of G for the topology of G. Proof. In view of Lemma 10.3.7 we need only show that, if U is an open neighborhood of the identity in G, then there exist some e > 0 and some compact set KeG such that N{K,£) c U. To this end let WI eWe G be symmetric open neighborhoods of the identity with compact closure such that W+ Wc U and cl{W l + WI)

..

c W. Let g = c{Xw I • Xw1), where Xwl denotes the characteristic function of WI and c > 0 is so chosen that IIg1l 2 :: 1. Then it is readily verified that g E C (G), get) > 0, t ~ G, and the supc port of g is contained in W. Furthermore, if s ~ U, then the supports of g and T (g) are disjoint, whence s

=2 > Consequently, if

5

E G and

1.

IIg - Ts {g)1I 2 ~ 1,

then

s E U.

Next, recalling the proof of Theorem 4.7.2 and the comments following it, we choose a nonnegative u ~ LI{G), lIulll = 1, such that I/u. g - gll2 < 1/3. Since the norm in L2 (G) is translation invariant, we have lIu • Ts(g) - Ts (g)1I 2 :: UTs(u • g) - Ts (g)1I 2

= lIu

• g - g/l2

I

(s

<3

/lu. g - u • Ts (g)/l2 < 1/3,

whence we see that, if s E G and then IIg - Ts (g)/l2 < /lg +

U •

gll2

E G),

+

/lu • g - u • Ts (g)/l2

lIu • Ts(g) - Ts (g)/l2

10.5. The Pontryagin Duality Theorem

311

III

<-+-+333 = 1.

Hence, if s ~ G and lIu. g - u • Ts (g)1I 2 ~ 1/3, then s E U. Moreover, since u ~ Ll(G) can be considered as an element of the B·-algebra A (G), namely, u is identified with T, and the o u Gel'fand transform of T and the Fourier transform of u agree, u we see from Theorem 10.3.1 that, if s E G were such that luCy) - Ts (u) (y)l < 1/3, y E G, then

..

..

..

lIu • g - u • Ts (g)1I 2

= lIu

• g - Ts(u) • gll2

= lI[u - Ts(u)] • gll2

< lIu - Ts (u)lIllg11 2

.. - T (u) . II

= lIu

..

s

CD

..

Thus, by the previous observation, if s EGis such that luCy) - Ts (u) (y)1 < 1/3, y E G, then s E U.

..

. .

..

Now U ( C (G), and so there exists some compact set KeG .. 0 .. such that luCy)l < 1/6, y E G - K. Then, if s E N(K,1/3), we see that for y E K we have

.

..

..

..

luCY) - Ts(u) (y)1 = luCy) - (s,y)u(y)I =

1~(Y)lll - (s,y)l

< II - (s,y)1 1

as

lIu1l1

..

< 3'

and if y E 6 - K,

= 1;

.

.

then

-

luCy) - TsCu) (y)' = lu(Y)"l - (s,y)l

10. B·-Algebras

312 1

< 3' as

•

•

lu(y)1 < 1/6, y E G - K.

Therefore, if s E N(K,1/3), then • y (G, whence we conclude that s E U, which is what we set out to prove.

•

A

lu(y) - Ts(u) (y)1 < 1/3, that is, N(K,I/3) c U,

o

As already indicated, this theorem entails that the mapping A • • A a : G - (G) is a topological isomorphism of G onto a(G) C (G) • It is now not too difficult to show that a is surjective. Theorem 10.S.2 (Pontryagin Duality Theorem). Let G be a loA • cally compact Abelian topological group and let (G) denote the A A • dual group of G. If a: G - (G) is defined for each t E G by • then a is a topological isomorphism of a(t)(y) = (t,y), y E G, • • G onto (G). Proof. In view of the foregoing discussion it is sufficient A • to prove that a(G) = (G). To do this we first show that a(G) is A • a closed subset of (G). Indeed, let K E cl[a(G)] and let {a(t~») be a net in a(G) that converges to K. Since a is a topological isomorphism, we see, given a compact set U C G that has a nonempty interior containing the identity of G, that there exists some ~o such that t~ - tw E U, ~ > ~o and w ~ ~o. In particular, t~ - t~o E U, a >~oJ and so a(t p) ( a(t po ) + a(U) = a(tao.+.U)' a >Po. However, a(t po + U) is compact, and so closed, in (G), as a is a homeomorphis=". Hence k E a(t po + U) C a(G), and a(G) is a closed subset of (G) "

• • then a(G) is a proper closed subset of Now, if a(G) ~ (G), • • A (G) , whence, since LI(G! is a regular commuta!ive Banach algebra, there exists.some h E Ll(G), h ~ 0, such that h[a(t)] = 0, t € G, the symbol h here of course denoting the Fourier transform of h A • • as an element of LI(~). Considering LI(G) as a subspace of M(G), we deduce at once from the Riesz Representation Theorem for continuous

10.5. The Pontryagin Duality Theorem

313

• linear functionals on Co (G) [L, p. 109] and Theorem 10.3.1 the existence of some TEA (G) such that o

•

ja h(y)T(y)

d~(y) ~

o.

Furthermore, a second appeal to Theorem 10.3.1, the definition of Ao(G), and the fact that Cc(G) is norm dense in Ll(G) allows us to assert the existence of some f ( C (G) such that c

• Ia h(y)f(y)

d~(y) ~

o.

However, by Fubini's Theorem [Ry, p. 269], we see that

• Ia h(y)f(y)

d~(~)

= Sa

h(y)[IG f(t) (t,y) dA(t)] d~(y)

= IG

f(t)[ja h(y)Ct,y) d~(y)] dl(t)

= IG

f(t)[ja h(y) (a(t),y) d~(y)] d4(t)

=IG

f(t)h[aCt)] dl(t)

•

= 0, which contradicts the choice of f. Therefore aCG)

• • = (G),

and the theorem is proved.

o

It may not be apparent to the reader where we have used Plancherel's Theorem (Theorem 10.4.4) to prove the preceding result. The • use occurs in appealing to the regularity of LlCG) (Corollary 7.2.3), a fact whose validity was established by means of Plancherel's Theorem. In view of the Pontryagin Duality Theorem it is customary to identify a locally compact Abelian topological group G with its • • It is this identification that was alluded second dual group (G). • • • to in the discussion of ~rJ and L in Section 4.7. A number of interesting results are now easy consequences of the

314

10. B*-Algebras

Pontryagin Duality Theorem. The first of these establishes the converse of Corollary 4.7.5 and is an immediate consequence of it and the Pontryagin Duality Theorem. Corollary 10.5.1. Let G be a locally compact Abelian topological group. Then the following are equivalent: (1) G is compact (discrete) •

• is discrete (compact). (ii) G If G is a locally compact Abelian topological group and h E L2 (G), then we shall denote by ~ the element of L2 (G) defined almost everywhere by ~(t) = h(-t), t E G. Corollary 10.5.2. Let G be a locally compact Abelian topological group, let A be a given Haar measure on G, and let ~ be • so chosen that Plancherel's Theorem is valid. a Haar measure on G A _.. .. If f E L2 (G), then [(f)] c f, where denotes the Plancherel • respectively. transformation on L2 (G) and L2 (G), Proof. Since the Plancherel transformation is an isometry, it suffices to prove the result for f E Vo =V n LI(G) because, by Theorem 10.4.3, Vo is norm dense in L2 (G). However, on Vo the Plancherel and Fourier transforms coincide and the Inversion Theorem (Theorem 10.4.1) is valid; so that, if f E V, then, on identifyo .. A ing G with (G) , we see that A_A

[(f) ] 'Ct) = =

la f(-y) (t,y) •

la fey) (t,y) A

d~(y)

d~(y)

= fet)

Thus

A_A

[(f) ]1

=f,

f E V, o

and the corollary is proved.

(t E G).

o

In particular. we observe that, if G is a locally compact Abelian topological group. A is a given Haar measure on G, and

10.5. The Pontryagin Duality Theorem

315

..

~

is Haar measure on G so chosen that Plancherel's Theorem is • valid for L2 (G) , then Plancherel's Theorem also holds for L2 (G) with the Haar measure on (6) taken to be A. This makes sense as we can identify (6) with G. Moreover, the inverse of the Plancherel transformation from L2 (G) to L2 (G) is the transfor-

..

..

.

..

•

r.! ..

..

mation .from L2 (6) to L2 (G) defined by g a 19) , g ~ L2 (6) , where denotes the Plancherel transformation from L2 (G) to

L2 (G). Our final corollary entails that M(G), G being a locally compact Abelian topological group, is semisimple. Corollary 10.5.3. Let G be a locally compact Abelian topological group. If ~ E M(G) and

•

~(y)

then

~

= fG

= o.

(t,y) ~(t)

=0

•

(y E G),

.

..

Proof. Identifying (G) with G, we see that, if f E LI(G) , then f E C (G) and, by Fubini's Theorem [Ry, p. 269], we have o

fG

..

f(t) ~(t)

.

= fG[ia

f(y) (t,y) d~(y)] ~(t)

=ia

f(y)[jG (t,y) ~(t)] d~(y)

=ia

f(y)~(y) d~CY)

=

•

o.

However, since L (G) is a self-adjoint commutative Banach algebra, we know that LI is norm dense in Co(G) (see. for example. the discussion following Definition 5.1.1). Consequently from the Riesz Representation Theorem for continuous linear functionals on CoCG) [L, p. 109] we deduce that ~ = o. 0

(!)-

.

The semisimplicity of MeG) is evident from this result because each y E G determines a ca.plex hOllOllOrphism of MeG). However,

10. B*-Algebras

316

as indicated in Section 4.8, the maximal ideal space of M(G) in gen• eral properly contains C. 10.6. A Spectral Decomposition Theorem for Self-adjoint Continuous Linear Transformations. A central concern in the theory of continuous linear transformations on Hilbert spaces is the development of ways of expressing a given transformation in terms of a sum or integral involving a family of orthogonal projections indexed ~y the points in the spectrum of the transformation. A variety of such results, known generally as spectral decomposition theorems, is known (see, for example, [BaNr, pp. 433-521; DS 2 , pp. 887-902,11411222; He, pp. 251-258,272-287; L, pp. 462-470; N, pp. 248-253; RzNg, pp. 233 and 234,272-277,280-291,313-320]). In this section we shall apply the Commutative Gel'fand-Naimark Theorem to obtain such a theorem for self-adjoint continuous linear transformations. Let V = (V,<.,.}) be a Hilbert space over ~ and suppose T E L(V) is self-adjoint, that is, T* = T. We shall denote by Ar the closed commutative subalgebra of L(V) generated by T. In other words, ~ is the smallest closed subalgebra of LeV) that contains T and the identity transformation I. Since T is selfadjoint, it is evident that is a commutative B*-algebra with identity, and so from the Commutative Gel'fand-Naimark Theorem (Theorem 10.2.1) we see that the Gel'fand transformation on Ar is an isometric *-isomorphism from ~ onto C(a(Ar)). Moreover, from our discussion of finitely generated algebras in Section 4.5 we know that a(~) is homeomorppic to the joint spectrum of the generators of ~ (Theorem 4.5.1), whence we conclude that a CAr) is homeo-

Ar

morphic to aATCT). However, aAT(T) = aL(V) (T). Indeed, this follows at once from Corollary 2.3.1 because aAT(T) Cffi by Proposition 10.1.1(viii). Thus afAr) is homeomorphic to aLcY)(T) = aft). Combining this discussion with Theorem 4.5.1, we immediately obtain the following lemma:

10.6. A Spectral Decomposition Theorem

317

Lemma 10.6.1. Let V be a Hilbert space over C and let T E L(V) be self-adjoint. If ~ is the closed subalgebra of L(V) generated by T, then the Gel'fand transformation on Ar is an isometric *-isomorphism from ~ onto C(a(T)). Moreover, TeT) - = T, T E aCT). Now, as indicated above, we wish to show that T and even certain functions of T can be approximated by sums of orthogonal projections that is, by sums of self-adjoint P (L(V) for which 2 P = P. In order to accomplish this it is tempting to approximate • T in C(a(T)) by some suitably nice functions and then use the isometric nature of the Gel'fand t~ansformation to obtain elements of ~ that approximate T. However, the elements of Ay so obtained need not be orthogonal projections, and, more to the point, Ay need not contain any orthogonal projections at all. For instance, if P E.'r were an orthogonal projection, then, since p2 = P, we -2 = -P, and so P would take only the values zero and would have P • one. Thus the set (T 1 T E aCT), peT) = 1) and the set • (T 1 T E aCT), peT) = 0) would be, in general, nonempty disjoint open subsets of aCT), and aCT) would be disconnected. Consequently, if aCT) is connected, then Ar contains no nontrivial orthogonal projections. Nevertheless, the original idea will work if we replace C(a(T)) by a somewhat larger algebra and extend the inverse of the Gel'fand transformation from C(a(T)) to this larger algebra. The new algebra we require is the algebra of bounded Baire functions of the first type on aCT); that is, the algebra S(a(T)) of bounded functions on aCT) that are pointwise limits of sequences of functions in C(a(T)). Clearly S(a(T)) is a normed algebra under pointwise operations and the supremum norm. Our task is to extend the inverse Gel'fand transformation from C(a(T)) to S(a(T)) in such a way that the image of S(a(T)) in L(V) will contain sufficiently many orthogonal projections to allow us to approximate T in an appropriate manner. We shall make this latter assertion more precise in Theorem 10.6.1.

10. B*-Algebras

318

With this in mind, let us denote the inverse Gel'fand transformation by ~; that is, ~: C(a(T» - ~ is such a ~apping that, if f £ C(a(T», then ~(f) E ~ is such that ~(f) = f. Clearly , is an isometric *-isomorphism from C(O(T» onto Ar. Let x,y £ V and define Fxy (f) = (~(f)x,y), f E C(a(T». Apparently Fxy is a linear functional on C(a(T», and the estimate IFxy(f)1 < lI~(f)(x)IlIlYIl ~ 1I~(f) IIl1xllllyll = \jfll~lIxllllyll

(f E C(a(T»)

reveals that Fxy is a continuous linear functional on C(O(T». Hence, by the Riesz Representation Theorem for continuous linear functionals on C(a(T» [L, p. 109], there exists a unique bounded complex-valued regular Borel measure ~ xy in M(a(T» such that FXy(f) = JO(T) f(") ~xy(T)

(f £ C(o(T»)

and II~ xy II = l~ xy I (a(T» < - IIxlillyli. Moreover, it is easily verified that the measures so constructed are such that (a)

~x(y + z) : ~xy + ~xz

(b)

~(x + y)z

(c)

~(ax)y = ~xy

(d)

~x(ay)

= ~xz

+ ~yz

= ~xy

(x,y,z £ Vj a £ C).

Furthermore, since every Baire subset of aCT) [Ry, p. 302], we see, given X,y £ V, that

is a Borel set

Ia(T) geT) ~xy(T) exists for each g £ 8(0(T». Consequently, given x £ V and g £ 8(a(T» - C(a(T» , the equation

10.6. A Spectral Decomposition Theorem

319

(y £ V)

defines an antilinear functional on V. The antilinearity of G x follows from the preceding conditions (a) and (d). Moreover, IGx(Y) I < IIgltlJl~xyU

< IIgllCDUxllllyli

(y £ V)

shows that Gx is even continuous and that IIG xII < - UgilCDIIxli. It then follows easily from the Riesz Representation Theorem for continuous linear functionals on Hilbert spaces [L, p. 390] that there exists a unique element ~(g)(x) £ V such that Gx (y) = {~(g)(x),y). y E V, and 1I~{g)(x)1I = II Gxll < UgIICDllxll. Now, given g belonging to B(a{T» - C(a{T» , if we repeat the previous argument for each x E V, we obviously obtain a mapping ,(g) from V to V that maps x to ~(g)(x), x E V. The preceding conditions (b) and (c) show that ~(g) is linear, while the previous estimate of 1I~(g)(x)1I reveals that
cp is an algebra homomorphism. cp(g)* =
where g*(T) = i(Tf, T E aCT).

10. B*-Algebras

320

Proof. We shall content ourselves with proving part (i). A similar argument establishes part (ii), the details being left to the reader. We define ~ : B(a(T)) ~ L{V) as in the preceding discussion. To show that ~ is an algebra homomorphism we need only prove that ~(gh) = ~(g)~(h), g,h (B(a{T)). To this end we assume first that g,h ( B(a(T)) are nonnegative. Then there exist monotone increasing sequences (g) and (h) of nonnegative functions n n in CrafT)) that converge pointwise to g and h, respectively. Consequently for each k the sequence (gnhk) C CrofT)) is a monotone increasing sequence that converges pointwise to gh k , whence we deduce, via an application of the Lebesgue Dominated Convergence Theorem [Ry, p. 229], that <~(ghk)(x),y)

= faCT)

ghk(T) ~Xy(T)

= lim

JO(T) gnhk(T) d~Xy(T)

= lim

<~(gnhk)(x),y)

n

n

= lim n

Sa(T) gn(T) d~~(h ) (x)y(T)

= Ia(T)

k

geT)

d~~(h ) (x)y(T) k

= <~(g)[~(hk)(x)],y) = <~(g)~(hk)(x),y)

(X,y €

V),

as the inverse Gel'fand transformation on crafT)) is an algebra isomorphism. A similar argument mutatis mutandis combined with the preceding result then reveals that <~(gh)(x),y)

= Ia(T) gh(T)

d~xy(T)

= lim

Ja(T) ghk(T) d~Xy(T)

= lim

<~(ghk)(x),y)

k

k

10.6. A Spectral Decomposition Theorem

321

= lim

<~(g)~(~)(x),y)

= lim

<~(hk)(x),~(g)*(y)

= l~m

IO(T) hk(T) ~x~(g)*(y)(T)

k

k

= Ia(T) h(T) ~x~(g)*(y)(T)

= <~(h)(x)J~(g)*(y) (x,y E V). However, any g E 8(a(T)) can be written as g = gl - g2 + i(g3 - g4)' where gk E 8(0(T)) and gk is nonnegative, k = 1,2,3,4. Thus the foregoing argument shows that for any g,h E 8(0(T)) we have <~(gh)(x),y)

= <~(g)~(h)(x),y)

from which we conclude at once that ~(gh) Therefore ~ is an algebra homomorphism.

= ~(g)~(h),

(x,yEV), g,h £ B(a(T)).

o

We are now in a position to see how to approximate a self-adjoint T in L(V) by orthogonal projections. If T £ L(V) is self-adjoint, then, as noted before, aCT) c~ Furthermore, by Theorem 2.1.2, we have oCT) C [-IITII.IITII]. For each t E aCT) we set gt = Xu(T)

n [-IITII,t]'

where XE, as usual, denotes the characteristic function of E. Clearly gt can be considered as an element of B(o(T)), and so, by Lemma 10.6.2, ~(gt) = Pt E L(V). Moreover, since , is a *-homomorphism and g2 = g = g*, we see that p2 = P = P*; that t t t t t t is, Pt is an orthogonal projection. In this way we obtain a family of orthogonal projections {Pt I t E oCT)). Furthermore, if • h E C(aCT)) is such that hCT) = T, T £ o(T) , that is, h = T, then, given e > 0, a straightforward argument using the uniform continuity of h on aCT) reveals that there exist points tk £ o(T) ,

10. B·-Algebras

322 k = 0,1,2, ... ,n, for which

such that

-IITII

~ to < tl < ... < tn ~

IITII

and

n

lh(T) -

t tk[gt (T) - gt

(T)]l <

I

(T

E a(T».

k=1 k k-l The details are left to the reader. However, since by Lemma 10.6.2 ~ is a norm-decreasing homomorphism from B(a(T» to Lev), we conclude that n

t tk (Pt - Pt ) II < e, k=l k k-l as ~(gt) = Pt ' k = O,I,2, •.• ,n, and ~(h) = T. Obviously this approxim~tion r~mains valid if we add to the original set of t k , k = O,I,2, ••• ,n, and so we see that the limit of the approximating sums of orthogonal projections exists and is equal to T as the maximum of the differences tk - t k_l , k = 1,2, ••• ,n, tends to zero. From the analogous procedure for the definition of the Riemann-Stieltjes integral, we also agree to denote this limit as an integral; that is, we write liT -

T =

IO(T)

t dP t .

Naturally, the integral notation is just a symbol for the previously described limiting process. Clearly the same argument is valid if we replace h by any f £ C(a(T» since each such f is uniformly continuous on aCT). We summarize this development in the next theorem. Theorem 10.6.1. Let V be a Hilbert space over C and let T E LCV) be self-adjoint. Then there exists a family of orthogonal projections {Pt I t E a(T)J contained in L(V) such that f(T) = I oCT ) f(t) dP t

(f £

C(a(T»).

A spectral decomposition theorem for T £ Lev) that are normal, that is, such that 11* = T*T, can also be obtained by means of the Commutative Gel'fand-Naimark Theorem. For further details the reader is referred to [HOI' pp. 96-100].

10.7. Almost Periodic Functions

323

10.7. Almost Periodic Functions. The notion of an almost periodic function was introduced in Section 1.2. We remind the reader that, if G is a locally compact Abelian topological group, then f E C(G) is said to be almost periodic if the set {Ts(f) I s E G) has compact closure in C{G), where, as usual, Ts(f)(t) = f(t - s), t E G. Equivalently, f is almost periodic if {Ts(f) I s E G] is a totally bounded subset of C(G); that is, given e > 0, there exist sl,s2, ••• ,sn in G such that for each s E G we have liT (f) - T (flU < e for some k = 1,2, •.. ,n. As alreafly indicated, s sk CD the set of almost periodic functions on a locally compact Abelian topological group G, denoted by AP(G), is a closed subalgebra of C(G) that contains the constant functions and so is a commutative Banach algebra with identity. Moreover, it is easily seen that, if f E AP(G), then so also is f, where f(t) = f(t), t € G. The verifications of these assertions are left to the reader. It is now evident that AP(G) is a commutative B·-algebra with identity, the involution being the obvious one of complex conjugation. Our goal in this final section is to use the Commutative Gel'fand-Naimark Theorem to obtain several results about the algebra AP(G) and its compact maximal ideal space A(AP(G)). In particular, we shall see that AP(G) is precisely the closure in C(G) of the algebra of trigonometric polynomials on G, that is, of the algebra of finite linear combinations of the continuous characters on G, and that A(AP(G)) is a compactification of G that can be viewed as a compact Abelian topological group. In the interests of completeness we make the following definition: Definition 10.7.1. Let compact Hausdorff topological cation of X if there exists • : X -Y such that t(X) is

X be a topological space. Then a space Y is said to be a compactifia continuous injective mapping dense in Y.

The reader should note that this is not the usual topological definition of a compactification (see, for example, [W 2 , pp. 151 and 152]), but it is the one appropriate to the discussion of this section.

324

10. B*-Algebras

The one-point compactification of a locally compact Hausdorff topological space, which we have had occasion to use at various points in the preceding sections, is a simple example of a compactification. We shall shortly examine one that is considerably more complicated. Our introductory remarks about AP(G) and the Commutative Gel'fand-Naimark Theorem (Theorem 10.2.1) combine to yield the next theorem. Theorem 10.7.1. Let G be a locally compact Abelian topological group. Then the Gel'fand transformation on AP(G) is an isometric *-isomorphism of AP(G) onto C[~(AP(G))]. Now in the usual manner we see that each point t (G defines a complex homomorphism of AP(G), namely, the complex homomorphism Tt defined by TtCf) = f(t), f (AP(G). Clearly the mapping t(t) = Tt , t (G, maps G into 6(AP(G)). We claim that • is actually continuous and injective and t(G) is dense in 6(AP(G)), whence we can conclude that ~(AP(G)) is a compactification of G. Before we can establish this, however, we need to prove a si~ple lemma. We shall denote by T(G) the algebra of all trigonometric polynomials on a locally compact Abelian topological group G. Lemma 10.7.1. group. Then T(G)

Let G be a locally compact Abelian topological C AP(G).

Proof. It obviously suffices to prove that (.,y) £ AP(G) for • each y E G. But if s (G, then Ts[(·'Y)] = (s,y)(·,y), and so {Ts[("Y)] , s E G] is a subset of {aC·,Y) , a E fl. The latter set, however, is compact in C(G), being the continuous image of f under the mapping a ~ a(·,y), a £ f. Thus (.,y) is almost peri• and consequently T(G) C AP(G). odic for each y E G, o Theorem 10.7.2. Let G be a locally compact Abelian topological group. Then ~(AP(G)) is a compactification of G.

10.7. Almost Periodic Functions

325

Proof. We need on}y show that the mapping • : G - a(AP(G)) defined above is a continuous injective mapping such that V(G) is dense in aCAPCG)). The continuity of • follows at once from the definition of V and the fact that the topology on a(AP(G)) is the relative weak* topology. If t,s E G and vet) = ,(5), then, since T(G).C AP(G), we see that (t,y) = Tt!Co,y)] = Ts[(·'Y)] = (s,y), y (G. However, by Corollary 4.7.3, G separates the points of G, whence we conclude that t = s. Hence • is injective. Finally, if ,(G) is not dense in a(AP(G)), then cl[V(G)] is a proper closed subset of a(AP(G)). Thus, since C[a(AP(G))] is a regular commutative Banach algebra (or, equivalently, by Urysohn's Lemma [W 2 , p. 55]), there exists some h E C[a(AP(G))] such that h ~ 0 but h[V(t)] = 0, t E G. But by Theorem 10.7.1 there • exists some f € AP(G) such that f = h. Moreover, (t

E G),

from which we see immediately that h = •f = 0, contrary to the choice of h. Thus ,(G) is dense in a(AP(G)). and a(AP(G)) a compactification of G.

is

o

It is important to note that the mapping • : G - a(AP(G)) is not in general a homeomorphism unless G is compact. In the latter case the assertion is a well-known fact of general topology [W Z' p. 83]. Actually a good deal more can be said about a(AP(G)) than just that it is a compactification of G. Indeed, one can extend the group structure of G to all of ~(AP(G)) in such a way that a(AP(G)) becomes a compact Abelian topological group. It is apparent how to make t(G) into a group. Namely, one defines yet) + '(5) = ,Ct + 5), t,s (G. It is easlily verified that this definition does make t(G) into an Abelian group, and it also, incidentally, makes • a group isomorphism. It is not quite so clear, however, how one should define T + W for arbitrary T,~ E a(AP(G)).

326

10. B*-Algebras

To see how this can be done we let T,w E 6(AP(G)) and let (U ) a and (W) be families of open neighborhoods of T and w, respeca tively, which are directed by inclusion and such that n u = (T) aa and n w = (w). For each a let aa

= (,(t)

E a

+

t(w) I t,w E G, t(t) E U , ,(w) E W ). a a

It is easily seen that this family of sets has the finite-intersection property, and hence, since 6(AP(G)) is compact, we see that

nacl(Ea )

We claim that n cl(E) is a single point. Indeed, a a suppose KI ,K 2 E nacl(Ea ). Since the topology on ~(AP(G)) is the relative weak* topology, to show that KI = K2 it suffices to prove that KI(f) = K2 (f), f E AP(G). So let f E AP(G) be given and suppose £ > o. Since f is almost periodic, there exist ~~.

sl,s2, ... ,sn in G such that for any s E G we have liT (f) - T (f) Ii < £/18 for some k = 1,2, ... ,n. Let U(T) = s sk Q) U(T;S/36;f O,f l , ..• ,fn ) be the weak* open neighborhood of T determined by fO = f, fk = TSk(f), k = 1,2, ... ,n, and £/36. That is, v E UeT) if and only if Iv(fk) - T(fk)l < 6/36, k = 0,1,2, ... ,n. Similarly we set U(w) = U(w;6/36;fo,f l , ..• ,fn ). Clearly, if t,s E G and either

'Ct),,(s) E U(T)

t(t),t(s) E U(w),

or

Ck Furthermore, we claim that, if then Irtct)

+

tCw)] (f) - r,Cs)

+

t(t),,(s) E U(T)

tCv)](f)1 = If(t

+ w) -

f(s

+

v)l < If(t

+

w) - f(t

= 0,1,2, ... ,n).

and

t(w),,(v) E U(w),

+ w) - f(s +

v)l

IIT_t(f) - TSj (f) Ii CD < s/18

To see this let Sj and sk be such that and IIT_vCf) - Tsk (f) II Q) < £/18. Then If(t

then

+

v)1

10.7. Almost Periodic Functions +

327

If(t

+

v) - f(s

= IT_t (f)(w) +

v)1

+

- T_t(f) (v) 1

1T -v (f) (t)

- T

-v (f) (s) I

< IT_t(f)(w) - Ts. (f) (w)1 + IT

s.

J (f)(w) - T

J

+

(f) (v)

s.

1

J

ITs. (f) (v) - T_t(f) (v)l J

+

1T

+

IT

+

IT

-v Sk sk

(f) (t) - T (f) (t) - T

(f)(s) - T

(f) (t) I

sk

(f) (s)

Sk

1

(f)(s)1

-v

< II T_t (f) - T5 . (f) 1\0)

+

lfj(w) - f j (v)1

J +

IITs. (f) - T_ t (f)II CD J

+ liT

+

-v (f)

- T

sk

(f) II

lfk(t) - fk(S)l

CD

+

liTsk (f)

- T (f) \I -v CD

e

<3

by the previous estimates. Consequently frcm the definition of the E we see that there ex exist t(t),'(s) E U(T) and t(w),.(v) E U(w) such that

and

whence we deduce that

10. B*-Algebras

328

IKl(f) - K2 (f)1 ~ IKl(f) - [y(t)

+

,(w)] (f)1

rt (s)

+

1£. (t)

+

Ir,(s) + ;(v)] (f) - K2 (f)1

e

e

+

t (w)] (f)

-

+ ,(v)] (f)

I

e

<-+-+-

333

= e. Since e > 0 is arbitrary, we have

KI(f)

= K2 (f),

and so

KI

= K2 .

Hence n cl(E) is a single point, call it K. We define Ot a T + w = x. The foregoing arguments show that T + w is uniquely defined, and they also entail that the operation of addition so defined is continuous in the Gel'fand topology on A(AP(G)). A similar development shows that inverses exist and that the operation of inversion is also continuous with respect to the Gel'fand topology. The details are left to the reader. In this way one extends the group structure of G to all of A(AP(G)), and with this group structure on A(AP(G)) topological group.

we see that

A(AP(G))

is a compact Abelian

We include the results of the preceding develppment in the next theorem. Theorem 10.7.3. Let G be a locally compact Abelian topological group. Then A(AP(G)) is a compactification of G, and the group structure of G can be extended to A(AP(G)) in such a way that A(AP(G)) becomes a compact Abelian topological group. Moreover, suppose H is any compact Abelian topological group and ~ : G - H is a continuous isomorphism such that (i)

H is a compactification of G.

(ii) The mapping ~*, defined by ~*(h)(t) = hr~(t)], t € G and h (C(B), is an algebra isomorphism of C(H) onto AP(G). Then H is topologically isomorphic to A(AP(G)J.

10.7. Almost Periodic Functions

329

Proof. We need only prove the last assertion of the theorem. Suppose H and ~ : G - H are as described. The mapping t : G - A(AP{G» defined above has the same properties as ~, and so we see that (~*)-lo'* is an algebra isomorphism of C[A(AP(G»] onto C(H). Consequently, by Theorem 4.1.4, Hand A(AP{G» are homeomorphic. Then, if ~ : H - A(AP(G» is a homeomorphism of H onto A(AP(G», we see at once that ~ is an extension of the group isomorphism • 0 ~-l : ~(G) - ,(G). Since ~(G) and t(G) are dense in H and A(AP(G», respectively, we conclude that ~ is a group isomorphism. Therefore Hand A(AP(G» are topologically isomorphic.

o

Considered as a compact Abelian topological group, A(AP(G» is generally called the Bohr or almost periodic compactification of G, and it is often denoted by bG. We shall do this in the remainder of this section in order to emphasize that bG is a topological group rather than just the maximal ideal space of AP(G). In view of the remark following Theorem 10.7.2 we see that, if G is a compact Abelian topological group, then bG is topologically isomorphic to G. The following lemma is now easily proved: Lemma 10.7.2. group. Then reG)

G be a locally compact Abelian topological .. =Letr(bG).

• then Proof. Obviously it suffices to show that, if y (G, (.,y) is a continuous character of bG and that every such character of bG is obtained in this fashion. Of course, (.,y) de• notes the Gel'fand transform of (.,y). But if Y E G and t,s E G, then

..

..

( •• y)

• [,(t)

+ '(5)] = (t +

s,y)

= (t,y)(s,y)

= (.,y) • rt(t)](.,y) • ['(5)],

10. B*-Algebras

330

where , : G - bG is as before. Since ,(G) is dense in bG and (.,y) • is continuous, we conclude that (.,y) • is a continuous character of bG. Conversely, suppose h E C(bG) is a continuous character of bG. By Theorem 10.7.1 there exists a unique f E AP(G) such that •f = h. Moreover, if t,s (G, then f(t

+ 5) = y(t + 5) (f)

•f[,(t)

+

,(5)]

= h[i(t)

+

,(5)]

=

= h[,Ct)]h[,(s)] = f(t)f(s), whence we see at once that f is a continuous character of G. Thus • • there exists some y E G such that (.,y) = f and (.,y) = h. C Since, by Corollary 4.7.3, the dual of a locally compact Abelian topological group G separates the points of G, an application of the Stone-Weierstrass Theorem [L, p. 333] reveals that reG) is norm dense in C(G) whenever G is a compact Abelian topological group. Combining this observation with Lemma 10.7.2 and the fact that the Gel'fand transformation on AP(G) is an isometry, we obtain the following theorem and corollary: Theorem 10.7.4. Let G be a locally compact Abelian topological group. Then reG) is norm dense in AP(G). Corollary 10.7.1. Let G be a compact Abelian topological group. Then AP(G) = C(G). Another application of Lemma 10.7.2 yields the next result. Theorem 10.7.S. Let G be a locally compact Abelian topologi• • cal group and let Gd deno!e the algebraic group G with the.discrete topology. Then (bG) is topologically isomorphic to Gd ·

10.7. Almost Periodic Functions

331

Proof. From Corollary 10.5.1 we know that (bG) • is a discrete group, so to prove the theorem we need only show that there • • exists a group isomorphism of Gd onto (bG) • The proof of Lemma 10.7.2, however, shows that the Gel'fand transformation on AP(G) provides such an isomorphism.

o

•

Replacing G by G in this theorem and applying the Pontryagin Duality Theorem (Theorem 10.5.2), we obtain the next corollary, the details being left to the reader. Corollary 10.7.2. Let G be a locally compact Abelian topological group and let Gd denote the algebraic group G with the discrete topology. Then bG• is topologically isomorphic to (G d ) • . A moment's reflection reveals that this corollary has the following significance: If G is a locally compact Abelian topological • is the group of the continuous characters of G, group, and G • then bG is the group of all the characters of G, that is, of all the bounded functions h on G such that h(t + s) = h(t)h(s), • = ~ we see that s,t E G. For example, if G = R, then, since R HR is the group of characters of i t This observation leads to the following approximation result for characters: Corollary 10.7.3. Let G be a locally compact Abelian topological group and let h be a character of G. If s > 0 and • such that t l .t 2•... ,tn are in G, then there exists some y E G Ih(t k ) - (tk,y)l < I, k = 1,2, ... ,n. Proof. By Corollary 10.7.2 we can consider h as an element • Since each tk (G defines the almost periodic function of bG. • • (t k ,·) E AP(G), k = 1,2, ... ,n, we deduce, on recalling that bG = • • ~(AP(G» and the definition of the Gel'fand topology on A(AP(G»). that (g

I

• Ih(t ) k

g E bG,

g(tk)1 < s, k

= 1.2 ••.. ,n)

10. B*-Algebras

332

•

defines an open neighborhood of h in bG. Consequently, on iden• for • with tCG) •C• tifying G bG, we see that there is some y E G • which Ih{t k ) - (tktY)I < 't k = 1,2, .•. ,n, because G is dense • in bG.

o

This corollary is actually an abstract version of the classical Kronecker approximation theorem, but we shall not pursue this observation since it requires more detailed information about the Bohr compactification of specific groups than we have at our disposal. For this and other general discussions of almost periodic functions and the Bohr compactification we refer the reader to [Bo; BR I , pp. 245-261, 430-438; HR 2 , pp. 310-312; Lo, pp. 165-173; Ru 1 , pp. 30-32; We, pp. 130-139].

REFERENCES

[A]

L.V. Ahlfors, Complex Analysis, 2nd ed., McGraw-Hill, New York, 1966.

[Ba]

G. Bachman, Elements of Abstract Harmonic Analysis, Academic Press, New York, 1964:-

[BaNr]

G. Bachman and L. Narici, Functional Analysis, Academic Press, New York, 1966.

[Bo]

H. Bohr, Almost Periodic Functions, Chelsea, New York, 1951.

[B]

A. Browder, Introduction to Function Algebras, Benjamin, New York, 1969. N. Dunford and J. T. Schwartz, Linear Operators, Part General Theory, Interscience, New York, 1958.

!:

N. Dunford and J. T. Schwartz, Linear Operators, Part II: Spectral Theory, Interscience, New York, 1963. (OkRa]

C.F. Dunkl and O. E. Ramirez, Topics in Harmonic Analysis, Appleton-Century-Crofts, New York, 1971. R.E. Edwards, Functional Analysis:Theory and Applications, Holt, Rinehart, and Winston, New York, 1965.

R.E. Edwards, Fourier Series: A Modern Introduction, Vol. Holt, Rinehart, and Winston, New York, 1967. [Ga]

I,

T. Gamelin, Uniform Algebras, Prentice-Hall, Englewood Cliffs, N.J., 1969.

[GRk~]

I.M. Gel'fand, O.A. Raikov, and G.E. ~ilov, "Commutative Normed Rings", American Mathematical Society Translations (2)~,

[Go]

115-220(1957).

R.L. Goodstein, Complex Functions, McGraw-Hill, New York, 1965.

[He]

G. Helmberg.

~

Introduction

~

Spectral Theory in Hilbert

Space, North Holland, Amsterdam, 1969.

333

334

[H]

References

E. Hewitt, "A survey of abstract harmonic analysis", Surveys in Applied Mathematics: Some Aspects of Analysis and Probability. Vol. !. Wiley. New York. 1958. pp. 105-168. E. Hewitt and K.A. Ross, Abstract Harmonic Analysis, Vol. Springer-Verlag, Berlin-GHttingen-Heidelberg, 1963.

!,

E. Hewitt and K.A. Ross, Abstract Harmonic Analysis, Vol. II, Springer-Verlag, New York-Heidelberg-Berlin, 1970. [Hy]

H. Heyer, Dualitlt lokalkompakter Gruppen, Lecture Notes in Mathematics, No. 150, Springer-Verlag, Berlin-HeidelbergNew York, 1970.

[HIP]

E. Hille and R.S. Phillips, Functional Analysis and Semigroups, American Mathematical Society Colloquium Publication XXXI, American Mathematical Society, Providence, R.I., 1957. K. Hoffman, Fundamentals of Banach Algebras, Instituto de Matematica da Universidade-do Parana, Curitiba, Brazil, 1962. K. Hoffman, Banach sraces of Analytic Functions, PrenticeHall, Englewood Cli~s, N.J., 1962.

[Hr]

L. HHrmander, An Introduction to Complex Analysis in Several Variables, D. Van Nostrand, Princeton, N.J., 1966.

[J]

B.E. Johnson, "The uniqueness of the (complete) norm topology", Bulletin of the American Mathematical Society,73,537-539(1967).

[Ka]

Y. Katznelson, An Introduction to Harmonic Analysis, Wiley, New York, 1968.-----

[L]

R. Larsen, Functional Analysis: An Introduction, Dekker, New York, 1973.

[Lb]

G.M. Leibowitz, Lectures on Complex FURction Algebras, Scott, Foresman, Glenview, Ill., 1970.

[LeRd]

N. Levinson and R.M. Redheffer, Complex Variables, HoldenDay, San Francisco, 1970.

[Lo]

L. Loomis, An Introduction to Abstract Harmonic Analysis, D. Van Nostrand, Princeton,-W.J., 1953.

[Nb]

L. Nachbin, The Haar Integral, D. Van Nostrand, Princeton, N.J. J 1965.

[N]

M.A. Naimark, Normed Rings, Noordhoff, Groningen, The Netherlands, 1964.

References

335

[Ph]

R.R. Phelps, Lectures ~ Choquet's Theorem, D. Van Nostrand, Princeton, N.J., 1966.

[Pi]

H.R. Pitt, Tauberian Theorems, Oxford University Press, London, 1958.

[Po]

L. Pontryagin, Topological Groups, Princeton University Press, Princeton, N.J., 1958.

[Ri]

C.E. Rickart, General Theory of Banach Algebras, D. Van Nostrand, Princeton, N.J., 1960.

[RzNg]

F. Riesz and B. Sz-Nagy, Functional Analysis, Ungar, New York, 1955.

[Ry]

H. Royden, Real Analysis, 2nd ed., Macmillan, New York, 1966.

[Ru l ]

W. Rudin, Fourier Analysis 1962.

~

Groups, Interscience. New York,

W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966. [5]

E.L. Stout, The Theory of Uniform Algebras, Bogden and Quigley, Tarrytown-on-Hudson, N.Y., 1971.

[Wa]

J.K. Wang, Lectures ~ Banach Algebras, Lecture Notes, Department of Mathematics, Yale University, New Haven, Conn., 1965.

[We]

A. Weil, L'integration dans les groupes topologiques et ~ applications, Actualites scientifiques et Industrielles 8690-1145, Hermann, Paris, 1951. J. Wermer, "Banach Algebras and Analytic Functions", Advances in Mathematics, Vol. 1, fasc. 1, Academic Press, New York, 1961. J. Wermer, Banach Algebras and Several Complex Variables, Markham, Chicago, 1971.

[Wr l ]

N. Wiener, The Fourier Integral and Certain of Its Applications, Dover, New York, 1958. N. Wiener, Generalized Harmonic Analysis and Tauberian Theo~, M.I.T. Press, Cambridge, Mass., 1964. A. Wilansky, Functional Analysis, Blaisdell. New York, 1964. A. Wilansky, TOpology for Analysis, Ginn, Waltham, Mass., 1970. J.H. Williamson,"Remarks on the Plancherel and Pontryagin theorems", Topology, 1,73-80(1967).

INDEX of absolutely convergent Fourier series, 22 radical, 82 separating function, 220 uniform, 255

A

..

A, 74

Ar,

316

A[e], 9 A(K),

17

AP(G), 18

almost periodic compactification, 329 almost periodic function. 18,323 apPlications of boundaries, 243-249

A(x), 150

A ~ (0), assumed after E!&! 4

approximate identity, 109 in LI(G), 109, 188

A_I' 29

Arens' Theorem, 48

A-I, 29

Arens-Royden Theorem, 158

connected component of the identity in, 156,158 B

AC(r), 17

Gel'fand representation of, 102 maximal ideal space of, 102 bG, 329 ~ilov boundary for, 228 bdy(E), 51 A(D), 17 8(a(T)), 317 Bishop boundary for, 228 Gel'fand representation of, 97 is not regular, 167-168 B*-algebra, 273 maximal ideal space of, 97 examples of, 273-275,282,323 ~ilov boundary for, 228 Banach algebra, 4 commutative semisimple, 81 adjoining an identity, 10 conditions to have an identity, algebra, 3 153-154,178 Banach, 4 finitely generated, 98 normal commutative. 176 commutative, 3 division, 35 regular commutative, 166 normed, 4 self-adjoint commutative, 132 337

Index

338

Tauberian commutative, 183 with involution, 273 belong locally to I, point, 201 at infinity, 201

at a

C(V) , 23

XA, 242 XE ' 93-94

C(X), 16 Beurling-Gel'fand Theorem, 81 Bishop boundary when X is compact metric, 227-228 Bishop boundary, example of nonChoquet boundary for, 242 existence of, 230-232 closed ideals in, 195 existence for a uniform algeexample of pC(X) = ~, 230-232 bra, 268 Gel'fand representation of, for A(D), 228 87-88 for C(X), X compact metric, maximal ideal space of, 86,90 227-228 is regular, 167 for a commutative Banach algesets of spectral synthesis for, bra, 226 196 for a separating function alge§ilov boundary for, 227 bra, 221 C (X), 16 o Gel'fand representation of, 88 Bishop-deLeeuw Theorem, 236-237 is regular, 167 Bohr compactification, 329 maximal ideal space of, 88 uniqueness of, 328 ~ilov boundary for, 227 boundary, applications of, 243-249 Bishop, 221 Choquet, 242 existence of, 242 for a commutative Banach algebra, 226 for a separating function algebra, 221 ~ilov, 225 existence of, 222

Cn ( [ a , b] ), 17

Gel'fand representation of, 92 maximal ideal space of, 92 ~ilov boundary for, 228 C*-algebra, 275 character, 331 continuous, 113

C, 3

characterization of singular elements, in C(X), 43 in commutative Banach algebras, 81 in self-adjoint semisimple commutative Banach algebras, 141

Cc(X), 16

Choquet-Bishop-deLeeuw Theorem, 271

CAo(G), 286

Choquet boundary, characterization of, 257,263 existence of, 242 for A(D), 242 for C(X), 242

C

cl(I), cl(E), 34,162 co(E), 233 co(E), 233

Index

339

for separating function algebras, 242

Oitkin's Theorem, 206 division algebra, 35

closed convex hull, 233 dual group, 127 closed ideals in C(X), 195 in L1 (G), G compact, 197

E

closure operation, 162 commutative algebra, 3

E, 162

Commutative Gel'fand-Naimark Theorem, 277

1}, 168 1}(x), 46

compactification, 323 almost periodic, 329 Bohr, 329 Stone-~ech, 90

ext(E), 233

complex homomorphism, 67

extreme point, 233

exp, 156

connected component, 15~1 of the identity in A , 156,158 continuity of inversion, 31 of quasi-inversion, 29 continuous character, 113 contour, regular, 143 spectral, 144 convex hull, 233 closed, 233

F

1, 105 o

f, 287

•f, 21,114,169,307-308 •f(k), 21 f

convolution, 19-20

* g, 19

IIfllAc'

22

IIflln, 18 D

IIfllp'

19

0, 17

IIfIlCD, 16,18

()A, 225

finitely generated Banach algebra, 98

A(A), 69 assumed non empty after E!&! 78 finitely generated commutative Banach algebra, maximal ideal discrete topological group, 106 space of, 99 Gel'fand representation of, 99 Ditkin's condition, 204

340

Index

Fourier coefficient, 21 transformation, 21,115

homomorphism, complex, 67 *-, 277

Fourier-Stieltjes transform, 128

homomorphism space, 69

function, almost periodic, 18,323 hull, 160 slowly oscillating, 190 hull-kernel closure, 162 functions which operate, 151 topology, 163 G

..

I

G, 113

I , 11 e

r, 17,21 • 116 f,

10 (E), 182

int(K), 17

y , 121

o Gel'fand-Mazur Theorem, 35

ideal, 4 left, 4 maximal, 4 modular, 7 primary, 216 proper, 4 regular, 7 right, 4 two-sided, 4

Gel'fand representation, 76 of AC(f), 102 of A(D), 97 of C(X), 87-88 of Co(X), 88 of Cn([a,b]), 92 of a finitely generated commu- identity modulo I, 7 tative Banach algebra, 99 of L1(G), 114 inverse, 5 of L (X,S,~), 93 left, 5 right, 5 Ge1'fand Representation Theorem, 74 Inversion Theorem, 298,301 CD

Gel'fand topology, 71

invertible, 5

Ge1'fand transform, ;6

involution, 273

Gel'fand transformation, 76

isomorphism, topological, 131 J

H

hel), 160

Jo(E), 182

Haar measure, 18

joint spectrum, 98

Index

341

quasi-inverse, 13 topological zero divisor, 40 zero divisor, 40

K

k (E), 160

kernel, 160

linear functional, multiplicative, 67 positive, 259

Kronecker approximation theorem, 332

Local Maximum Modulus Theorem, 220

L

L(V) , 22

M , 12 e •~, 128

A (G), 282 o

II~II, 20

A, 18

l~L 20

Ll(G), closed ideals when G is compact, 197 closed translation-invariant linear subspace of, 186 condition to have an identity, 107 contains an approximate identity, 110,188 failure of spectral synthesis when G is noncompact, 199 Gel'fand representation of, 114 is regular, 173 is semisimple, 117 is Tauberian, 184 maximal ideal space of, 114,126 satisfies Ditkin's condition, 214 sets of spectral synthesis, 215 sets of spectral synthesis when G is compact, 198 ~ilov boundary for, 228

~

Lp (G), 19

L (X,S ,toLL 18

~ Gel'fand representation of, 93

maximal ideal space of, 93 left ideal, 4 inverse, 5 modulus of integrity, 46

*

\I,

20

M(G), 20 is semisimple, 315 maximal ideal space of, 128 Malliavin's Theorem, 199 maximal ideal, 4 maximal ideal space, 69 of AC(r), 102 of A(D), 97 of C(X), 86,90 of Co(X), 88 of cn([a,b), 92 of a finitely generated commutative Banach algebra, 99 of LI(G), 114,126 of L (X,S,~), 93 of MTG), 128 maximum modulus set, 221 Maximum Modulus Theorem, 218-219 Mergelyan's Theorem, 101 modular ideal, 7

Index

342 modulus of integrity, 46 left, 46 right, 46

Polynomial Spectral Mapping Theorem, 57

Pontryagin Duality Theorem, 312 multiplicative linear functional, 67 positive linear functional, 259 primary ideal, 216 N

proper ideal, 4 N(K,e), 293,309-310

Q

nilpotent, 33 topological, 33 Noncommutative Gel'fand-Naimark Theorem, 280

quasi-inverse, 13 left, 13 right, 13

normal commutative Banach algebra,quasi-invertible, 13 176 quasi-regular, 13 normed algebra, 4 quasi-singular, 13 p

R P(K), 17 Parseval's Formula, 170,305

• 115-116

~

peak point, for a commutative Banach algebra, 226 R(K), 17 for a separating function algebra, 221 R(f), 55 weak, for a separating function algebra, 262 Rad(A), 82 peak set, for a separating function algebra, 262

pA, 221

radical, 82 perfect set, 205 radical algebra, 82 Plancherel's Theorem, 169,301, 307-308

regular commutative Banach algebra, 166 P1anchere1 transform, 169,305 contour, 143 curve, 143 Plancherel transformation, 169,305 simple, 143 simple closed, 143

Index

343

element, 5 ideal, 7 representing measure, 253 and the Choquet boundary, 257 and the §ilov boundary, 269-270 Riemann-Lebesgue Lemma, 114 right ideal, 4 inverse, 5 modulus of integrity, 46 quasi-inverse, 13 topological zero divisor, 40 zero divisor, 40

for C(X), 227 for Co(X), 227 for CO((a,b]), 228 for L (G), 228 for a !ommutative Banach algebra, 226 for a separating function algebra, 225 §ilov-Arens-Calderon Theorem, 151 simple closed regular curve, 143 regular curve, 143

a(x), 54

singular element, 5 in C(X), 43 in a commutative Banach algebra, 81 in a self-adjoint semisimple commutative Banach algebra, 141

*-homomorphism, 277

slowly oscillating function, 190

s

self-adjoint, 132,275 smooth are, 143 commutative Banach algebra, 132 element in a B·-algebra, 275 Spectral Mapping Theorem, 152 semisimple, 81

spectral radius, 58

separating function algebra, 220 Spectral Radius Formula, 58 set of spectral synthesis, 194

spectral radius norm, 130

for C(X), 196 for Ll (G), G compact, 198 spectral synthesis, failure in for L (G), 215 Ll(G) when G is noncompact, for a ~emisimple regular commu199 tative Banach algebra that in C(X), 196 satisfies Ditkin's condiin L1(G), G compact, 198 tion,208-209 in Ll (G), 2l~ in a commutatIve B·-algebra, set, of spectral synthesis, 194 278 in a semisimple regular commuperfect, 205 tative Banach algebra that satisfies Ditkin's condi§ilov boundary, 225 and representing measures, tion, 209 problem of, 194 269-270 characterization of, 225 set of, 194 existence of, 222 spectrum, 54,69 for AC (n, 228 joint, 98 for A(D), 228

Index

344 Stone-~ech compactification, 90

strong boundary point, 222,262 structure space, 69 suba1gebra, 4 superalgebra, 48 support of

~,

235

symmetric set, 275 T

IITII,

Ditkin, 206 R.E. Edwards, 38 Gel'fand, 23,74 Ge1'fand and Mazur, 35 Gel'fand and Naimark, 277,280 Ma11iavin, 199 Mazur, 40 Mergelyan, 101 Pitt, 191 Plancherel, 169,301,307-308 Pontryagin, 312 Riemann and Lebesgue, 114 ~ilov, 61,222 ~ilov, Arens, and Calderon, 151 Tauber, 189 Wermer, 255 Wiener, 104,190 Wiener and L~vy, 154

22

T , 46 x TX, 46

topological isomorphism, 131 . zero divisor, 40-41 in C(X), 43 in a separating function algebra, 246-247

Tf' 280 topologically nilpotent, 33 T (f), 18,104

s T(G) , 324

T,

66

totally disconnected, 95 transform, Fourier, 21,115 Ge1'fand, 76 P1ancherel, 169,305

T , 69-70

e

T , 70 CD

Tt , 86

(t,y), 114

transformation, Fourier, 21,115 Gel'fand, 76 order preserving, 93 P1anchere1, 169,305 translation invariant linear subspace, 186

Tauberian commutative Banach algebra, 183

translation operator, 18,104

Tauber's Theorem, 189

trigonometric polynomial, 187

theorem of Arens, 48 Arens and Royden, 158 Beurling and Gel'fand, 81 Bishop and deLeeuw, 236-237 Choquet, Bishop, and deLeeuw, 271

two-sided ideal, 4 topological zero divisor, 40 zero divisor, 40

Index

345

•X,

U

74

o

x, 174

U(K,c), 123

U(T;C;x l ,x 2, •.• ,xn), 71

x*, 139,273

uniform algebra, 255

IIxlla,

58

• lIxll.,

134

unique maximum point, 222

x

y, 13 ~(x), 46

v

0

z

vanish on an open set, 235

w

Z, 3

•

Z, 116

weak peak point, 262 Wermer's Maximality Theorem,

Z(x), 160 255

Wiener's Tauberian Theorem, 190 Wiener's Theorem, 104 Wiener-L:vy Theorem, 154

x x

-1

,6

Z (I), 160

zero-dimensional, 93 zero divisor, 40-41 left J 40 left topological, 40 right, 40 right topological, 40 topological, 40-41 two-sided, 40 two-sided topological, 40 zero set of an element, 160 of an ideal, 160

about the book . . .

This textbook i~ designed to provide an introduction to the theory of commutative Banach algebras. The book not only dea ls with abstract theory, but illustrates the usefuln ess of Banach algeb ras in the s tud y of harmonic analysis and function algehras as we I!. The first half of the book is primarily devoted to the general theo ry of Banach algeb ras. Many spec ific examples are included in this part to illustrate the princi pies discussed. The second half o f -the book cons iders more speciali zed '.opics, among which are: Wie ner's Tauberian Theorem; the problem of spectral synthes is ; th e Bishop. 'Choquet, and Silov boundaries; and Wermer's Maximality Theorem. The second part also examines th e Commutative Gel'fand-Naimark Theo rem , Planchere l's Theorem, th e Pontryagin Duality Theorem, almost periodic fun cli o ns. and th e Bohr com pacti fi Ci_lti on.

Bana('h Algebras i ~ a stimul at ing text ro r g raduate stude nts with a backg ro und in abs trac t algebra. ge neral w[',,:ogy, real and com plex analysis , and fun ctional analys is.

about the author .. . RONALD L ARSEN has been Associate Professor of Mathemati cs at Wcsicyan Universit y in Middletown, Connecticut. since 1970. He previously tau ght at the Un iversity of California al Sanla C ru z (1965- 70).a nJ ~t Yale Univers ity (1963- 65). He received hi s B.S .

(1957) and M.S. ( 1959) from Michigan State University and his Ph. D. (1964) from Stanford University .

Professor Larsen 's rcsC!arl:h interes ts center on harmo ni c analysis and Banach algebras. He is the author of thre,e books. The Multiplier Problem, Allllllroductioll to the Theory of Multipliers, and Functional Allal)'sis: All Iwroc/uction (Marcel D=k ker, In c.), and numerous articles. The author wa ~ the recipient of a Ful bright-Ha ys Act Advanced Research Grant to the University of Oslo ( 1968-69) . He is a mem ber of the Ameri ca n M.Hhematical Society. the Mathe mat ical Assoc iation of Am eri ca , an d the Norsk Malematisk Forening.

Prin(ed ill rltl' Ullired Sf{/Il:'.~ of AIII('rica

ISB N: 0 - 8247- 6078- 6

MARCEL DEKKER , INC., NEW YORK