Weakly Almost Periodic Functionals on the Fourier Algebrapeople.virginia.edu/~der/pdf/der29.pdf · Stieltjes transform 9) of M(G), the measure algebra of G, into W(d). If G is compact

Weakly Almost Periodic Functionals on the Fourier AlgebraAuthor(s): Charles F. Dunkl and Donald E. RamirezSource: Transactions of the American Mathematical Society, Vol. 185 (Nov., 1973), pp. 501-514Published by: American Mathematical SocietyStable URL: http://www.jstor.org/stable/1996454 .

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TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 185, November 1973

WEAKLY ALMOST PERIODIC FUNCTIONALS ON THE FOURIER ALGEBRA

BY

CHARLES F. DUNKL AND DONALD E. RAMIREZ

ABSTRACT. The theory of weakly almost periodic functionals on the Fourier algebra is herein developed. It is the extension of the theory of weakly almost periodic functions on locally compact abelian groups to the duals of compact groups. The complete direct product of a countable collection of nontrivial compact groups furnishes an important example for some of the constructions.

I. INTRODUCTION AND BACKGROUND

Introduction. Eberlein used ergodic methods to create weakly almost periodic functions on groups [5]. In this paper we present a nonabelian extension of the theory. Our extension is not in the already explored direction of functions on groups, but rather we use the idea that the space of bounded measurable functions on a locally compact abelian (l.c.a.) group G is the dual of the Fourier algebra of G (the dual (l.c.a.) group of 6), where the Fourier algebra of G is the set of Fourier transforms of the integrable functions on G. The Fourier algebra is defined on each locally compact group, and has been investigated by Eymard [6]. It is a commutative Banach algebra of continuous functions on G vanishing at infinity. Its dual turns out to be the von Neumann algebra generated by the left translations on the Hilbert space of square-integrable (left Haar measure) functions on the group. This algebra of operators is a module over the Fourier algebra, and we use module-type definitions of weakly almost periodic (w.a.p.) elements of it.

For locally compact group G with Fourier algebra A(G), let W(G) be the set of 4) E A(G)* (the dual of A(G)) such that the map f f-* f 4 of A(G) -* A(G)* is a weakly compact operator (f p is the module action). Many of the classical theorems on w.a.p. functions also hold in this setting of w.a.p. functionals on the Fourier algebra. For example, W(6) is a Banach space with a unique invariant (in a sense to be defined) mean, and there is a monomorphism (the Fourier- Stieltjes transform 9) of M(G), the measure algebra of G, into W(d).

If G is compact then the easily accessible structure of the dual of G allows a more detailed study. The dual of A(G) is an algebra of bounded matrix-valued functions on a discrete space, and we are able to use the idea of quasi-uniform

Received by the editors September 7, 1970. AMS (MOS) subject classifications (1970). Primary 43A10, 43-00, 43A75; Secondary 22D40,

43A07. Key words and phrases. Weakly almost periodic, almost periodic, means, quasi-uniform conver-

gence, Fourier algebra. Copyright i 1974, American Mathematical Society

501



502 C. F. DUNKL AND D. E. RAMIREZ

convergence (invented for use on scalar valued functions) to characterize weak compactness. This technique allows us to show that if G is the complete direct product of a countable collection of nontrivial compact groups, then 9M(G) is not dense in W(G), and W(ci) is a proper closed subspace of A(G)*.

Here are some details on the organization of this paper. It is split into sections and chapters.

?1 contains the present material and Chapter 1 dealing with the classical abelian theory from the A(G)-module point of view.

?11 is the main part of the paper, and deals with compact groups. Chapter 2 gives the basic definitions and properties of the w.a.p. functionals and the mean, and uses the ergodic semigroup theory of Eberlein [5]. Chapter 3 deals with quasi- uniform convergence, and heavily uses the theory of weak topologies on Banach spaces. In Chapter 4, the T-sets are constructed. These are "thin" subsets of the dual of G, which have the property that any 4) E A(G)* which is carried by one, is w.a.p. Chapters 5 and 6 show, for the complete direct product G of a countable collection of nontrivial compact groups, that 9M(G) is not dense in W(G) and that W(O) # A(G)*.

?111 contains Chapter 7 dealing with almost periodic functionals, and Chapter 8, which is a swift extension of the material of Chapter 2 and 7 to all locally compact groups.

Notation is cumulative throughout the paper. In ?I1, G denotes a compact group, except when, for purposes of illustration, we consider l.c.a. groups.

During the preparation of this paper the authors were partly supported by NSF grant GP-8981.

Chapter 1. Abelian background. In this section, we will let G denote a locally compact abelian (l.c.a.) group with G its dual group. An essentially bounded Borel function O on C is said to be weakly almost periodic (w.a.p.) if and only if the set of translates of 0, denoted by 0(,O), is relatively weakly compact in LX (G ), the space of all essentially bounded Borel functions on G (see [5]). We will first give an equivalent definition (for w.a.p. functions) which will allow us to develop the theory of w.a.p. functions on duals of nonabelian groups.

We let LI (G) be the Banach algebra of integrable functions on G with convolution * as the multiplication. The Fourier transform 9 takes LI (G) into C B(G ), the space of continuous bounded functions on G. The Fourier algebra of G, A(G), consists of those continuous functions f on G for which 9f E LI (G). The space A(G) is a Banach algebra under the pointwise operations and the norm lIf 11 = 119fllj. Now since A(G) _ L' (), the dual of A(G) can be identified with Le(G), and the pairing is given by <KfO> = Jf (9f)(y)O(y)dmO(y), f E A(G),

E Le (G ), and where m< is the Haar measure on G. We will now need the fact that Le (G ) is a Banach module (see [6], [7]) over

A(G): givenf E A(G), O E Li(G), definef E E LV(G() by <g,f 4)> = <fg, 4>,g E A(G).



WEAKLY ALMOST PERIODIC FUNCTIONALS 503

Fix ?J e L(G ). Let T be the bounded operator from A(G) to Le (G ) given by

fH-f f *,f E A(G ). Note If * 4|L < Ilf IIA 41. Now for g E A(G),

<g,f 4 )> = <fg,> = f 9(fg)(y)4)(y)dm0(y) = <g,(9f) * 4)>

where h(y) = h(-y), h E Le (G). Thus Tf = f * 4 = (9f )p * 0 which is in CB (). We now use the theory of the representation of operators with values in a commutative C*-algebra (see [4, p. 4901). The operator T induces an adjoint map r from I,3C the Stone-tech compactification of 6, into L(G) by the rule <f,(y)> = Tf(y), f E A(G), y E /G3. Thus for f E A(G), y E G C f80, we have <f,r(y)> = (J * 4+)(y) = <f,R(y)0> where (R(y)4')(z) = 41(y + z), Ap

e LX (G ), z E G. Hence r(y) = R(y)o for y E G. Now x: ,B- Le() is always continuous in the weak-* topology

o(L* (a), LI (s)). Further, T is a weakly compact operator if and only if r is continuous in the weak topology o(Le (0), L (G')*) (for X a normed space, X* denotes the dual of X). Now if T is a weakly compact operator, then it follows that 0(4)) is relatively weakly compact. Conversely, if 0(4) is relatively weakly compact, it follows by standard point-set topology methods that T is continuous into LX (G) with the weak topology. Thus we have shown the following theorem.

Theorem 1.1. Let G be a locally compact abelian group. For 4 E8 Lj'(G() to be weakly almost periodic it is necessary and sufficient that the map f i-+ f * 4 from A(G) to CB(G) (f * 4 = J * 4) is a weakly compact operator.

Remark 1.2. Choose an approximate identity {ux} in LI (G ) (G is an l.c.a. group) such that ul= ux and jlux II, = I for each X. Then for any 4) E LV(G) we have that ux 4) = u* 4 converges weak-* to 4. Each ux * E CuB(G), the space of uniformly continuous bounded functions on 6. Now suppose further that 4) is w.a.p.; then 0(4) is relatively weakly compact. Thus {ux 4}) has a weak cluster point in 0(4)) c CjB(Gc) (recall for convex sets, the norm closure is the weak closure), which must be 4. This proves the following:

Theorem 1.3. If 4 E8 L '(O) and is w.a.p., and G is l.c.a., then 4 is uniformly continuous.

II. COMPACT GROUPS

Chapter 2. Basic definitions and the mean. Let G be a compact nonabelian group. Using our previous notation [2, Chapter 7], we let 6 denote the set of equivalence classes of continuous unitary irreducible representations of 0. For a E G, let T, be an element of a. Then T7 is a homomorphism of G into U(na), the group of na X na unitary matrices where n, is the dimension of a. We use

T7(x)ij to denote the matrix entries of T(x), 1 < i, j < na, and Tm., to denote the function x -* T, (x)ij. Now T7 (xy)ij = I "' I Ta (x)ik T7 (y)k; and Ta (y'l),j = Furthermore, Taij E C(G), the space of continuous functions on G. For a E G,




let x.,(x) = trace(T.(x)) = T 7;(x)i,. The function is called the character of a and it is independent of the choice of T, in a.

Let X be an n-dimensional complex inner product space. Let 2B(X) be the space of linear maps from X -- X. We define the operator norm of A E .23(X) by I|A IL = sup{IAXI: t E X, |tI < 1). The trace of A, Tr A, is E (A&,) where { i}7=l is any orthonormal basis for X and (,) denotes the inner product in X. Let IAI denote (A*A)I/2. The value IAIL is the spectral radius of JAI; i.e. max{Xi: I ? i < n) where Xi are the eigenvalues of JAI.

Let p be a set {4-Oa a e G' where . E 2B(Cna)) such that sup{jlk. IL: a E G} < oo. The set of all such O is denoted by eP (G). It is a Banach algebra under the norm Il4IO1 = sup{lka IL: a E G) and coordinatewise operations. Let M(G) denote the measure algebra of G, that is, the space of finite regular Borel measures on G with convolution as the multiplication. For , e M(G), the Fourier-Stieltjes transform of t, js is a matrix-valued function defined for a E -

by a f Gi = f 7;(x-l)dA(x). Note that ,i E Z'(). We sometimes write 9,L for ii.

Let A E B(X) where X is a finite-dimensional complex inner product space. We define the dual norm to 11-II by IfAII, = sup{ITr(AB)I: IIBIO < 1}. This norm can be also characterized by IA II, = Tr(|AJ). For 0 E t?O(G), we put II411, = jm =, n I I II1. Let -l '(G') = {, E L` (G'): 1I11 < 001. The space XL(G) is a Banach space under the norm 11111. For e EC X(d), let Tr(4) = xE na Tr(O.).

We will now define A(G), the Fourier algbra of G. Let A(G) be the set of f E C(G) for which f E ?'(0). We norm A(G) by lif hA = IIl9fIIh-t =Jli

- E a jIn. < oo. Note that A(G) is isomorphic to -el(G') by f -+ 9f because for any 4) E /P(I) the function f(x) = Eaeo na Tr(4)h T,(x)) is in A(G); further,

lIfL = sup{J na Tr(4 Ta(X))f.x)| X ? = < falInaIIi = ll4)lil aEG= uE

We now recall the following facts (see [2, Chapter 8]): A(G) -' () by f -* 9fjf E A(G); A(G) is an algebra under pointwise multiplication; the dual of -I (G ) is 12 (G) and the correspondence is given by <A,+>4 Tr)(>), ) E ?' (O ), O E (O ). Thus the dual of A(G) can be identified with -'(G(i)

and the pairing is given by <Kf> = Tr(oJ), f E A(G), 4 E (G). The module action of ?c1(G) over A(G) is defined by the following: f E A(G), 4 E ?0(d), f * 0 E PO(d) by <g,f 4)> = <fg,)>, g E A(G). Note lIf -,OI < IIflIA 1II1IL

Definition 2.1. For G a compact group and 4 e 1t (GQi), we say that 4 is weakly almost periodic (w.a.p.) if and only if the map f t f - 4 from A(G) to -LI(Gc) is a weakly compact operator. The space of all such 4 is denoted by w(d).




We denote the unit ball in A(G) by B. We denote by P the set of continuous positive definite functions on G which at e, the identity of G, have the value 1. Note that P C A(G) and f E P if and only if f E B and fa. is a positive matrix for each a E G.

Proposition 2.2. For 4) E ?' (c), 4 E= W(G ) if and only if B .) = {f * 0: f E B} is relatively weakly compact in ?4 (G ) if and only if P*) = {(f 4 ): f E P} is relatively weakly compact in eL' (G ).

Proof. We need only observe that any hermitian operator is a difference of two positive definite operators. Thus B is contained in the balanced convex hull of 2P. O

Definition 2.3. For 4 E - (G ), we define 0(4)) to be the set P 4)O. (Observe that 0(4)) is convex.)

Remark 2A. For 4 E eP (O ), to show that 0(4) is relatively weakly compact it is necessary and sufficient to show that 0(4)) is relatively weakly sequentially compact; equivalently, 0(4) is relatively weakly countably compact (see [4, p. 430]).

Theorem 2.5. W(O) is a closed *-subspace of ?0e (O

Proof. For 4 E (G(), let T: A(G) -* (G(i) be defined as before by Tf = f* 4), f E A(G). Note that the operator norm of T, 11 TII, is the same as the sup norm of 4 in ?0' (G(), 1)IL. Thus to show that W(d) is a closed space we need only invoke [4, p. 4381: a uniform limit of weakly compact operators is weakly compact. The *-invariance follows since 0(4*) = 0(4))* and 4) 0 * is a weakly continuous map. a

Theorem 2.6. W(6) is a module over A(G).

Proof. Let 4 EG W(G) andf E A(G). We need to show thatf 4 E W(G). We observe that B - (f 4)) C (jlfjlAB) * 4. 0

Definition 2.7. Let at(d) be the closure of 9M(G) in 2*(Gc).

Theorem 2.8. The algebra .Al(O) is contained in W(O).

Proof. It will suffice to consider a positive measure ,u E M(G) with 11l41 =I. Let =# be the Hilbert space L2(G,j). Consider the map S: IC (G) by (g,Sf) = f (fg)^d,u, g E A(G), f E W. Now f(gSf )I < IIg9ILYlif IL < 11gILA

lif L1,, and thus S is strongly continuous and hence weakly continuous [4, p. 422]. Let {fnQ C A(G ), IIfnIIA < 1. Then 4fn) is contained in the unit ball of 4# which is weakly sequentially compact. Thus there is a subsequence fn -k h weakly in cJf. Thus Sfn ---k Sh weakly. It remains to observe that Sf = f* ji, f E A(G). This follows from:

(g, Sf) =f(fg) dA = Tr(ji(fg)^

(<fg,I>=<g,fiA>, f,gEA(G). 0



506 C. P. DUNKL AND D. E. RAMIREZ

Note that this implies that Co(d) C W(d), where e80(d)-(

E ?~"(G): II4a)It > 8 for only finitely many a E Gd for each e > 01, since o0(G) is the closure of 9L1 (G) (see [2, Chapter 8D. Recall that P = {f E A(G): f is positive definite with f(e) = 1). Observe that

P is a commutative semigroup of operators on e' (d). Definition 2.9. The semigroup P is called ergodic (see [5, p. 220D if it possesses

at least one system of almost invariant integrals. By such a system one means a family of transformations {PA} such that:

(1) pA is a linear transformation of .L' (G') into_itself, (2) for each 4 E L(G ) and each X, pA\ 4 E 0(4),

(3) supAIIp|II < 00,

(4) for each f E P, limj(fpx - Px)4) = 0 and limp(Pjf - Px) = 0, 4 e -,co (G').

Theorem 2.10. The semigroup P is ergodic.

Proof. Let (A} be a neighborhood basis for e E G. Let UA E P with support UA C A. We order the net {UA} by uA > U, if and only if A C K. Now since W(G) is an A(G)-module (Theorem 2.5), we define the linear transformation PA from W(d) to W(d) by pA4) = UA * 4, 4 E W(G). Properties 2.9(1), 2.9(2), and 2.9(3) are immediate.

To show property 2.9(4) it will suffice to show for f E P that jIfUA - UA IA -- 0. It is therefore enough to show that IIuXgjjI A* 0, g E A(G) with g(e) = 0. Now recall that each closed primary ideal in A(G) is maximal (see [6, p. 229]). Thus given g E A(G) with g(e) = 0 and e > 0, there is h E A(G) with h = 0 on a neighborhood V of e such that lg - hIlA < e. Now for A C V, we have that IIUgjjA < jjuu(g - h)IIA + IIuAhIIA < E 0

Theorem 2.11 To every 4 E W(G) there corresponds a constant denoted by 0(4) such that for 4, 4, E W(G ) one has

(1) 0(a4 + 4,) = a0(4)) + 0(4,) and I0(4)I < I I 1 (a E C), (2) 0(I ) = 1 (I denotes the identity operator in ZL0 (G)), (3) if 4)a > Ofor all a, then f(4)) 2 0, (4) UA 4* -*A 0(4))I in the norm of 2 (G'), {UAI as in 2.10, (5) 0(4*) = 0()), (6) g(f 4 )) = 0(4)f (e), f E A(G), (7) properties (1), (2), and (6) uniquely determine 0.

Proof. Since P is an ergodic semigroup (Theorem 2.10), the ergodic theory (see [5, p. 220]) yields that each 4) E W(G ) has a unique fixed point F(4)) in the weak closure of 0(4); i.e.,f * F(4)) = F(4), for eachf E P. We now argue that F(4) is a constant multiple of the identity operator I in e0 (G ).

For 4, E 2L (G ) we define spt 4 to be the intersection of all the compact sets K in G for which f 4 , = 0, f E A(G), whenever spt f (the usual support) is




contained in the complement of K. Further spt(f * 4) c sptf n spt 41, f E A(G), 4 E L'(G(). So if f * 4 = A, all f E P, then spt 4, C {e}. Thus for

4 E W(G), F(4) is a constant multiple of I and we write F(O) = _J(c)I (see [6,

Chapter 41). Part (4) is from [5, p. 220]. The other claims follow from a standard

argument. 0l

Theorem 2.12. For % E M(G), .c(f) =,

Proof. Let f E P, then f *f u = 9(Jd,). Thus as spt f -- {e}, f * -,({e})I .O

Chapter 3. Quasi-uniform convergence. For S a compact (Hausdorff) space, a

sequence in C(S) is weakly convergent if and only if it is bounded and quasi-

uniformly convergent on S (see [4, p. 269]). Also for S a locally compact space,

a bounded sequence {fj) from C8(S) converges weakly to some f E CB(S) if

and only fn -*" f pointwise on S and every subsequence of { fn} converges quasi-

uniformly on S to f (see [4, p. 281]). That quasi-uniform convergence is useful in

studying w.a.p. functions on l.c.a. groups has been observed in [9]. We will

develop similar results for ?X' (G ) (a compact group G). We will use the concept

of quasi-uniform convergence which was introduced by Arzela in 1884 for the

unit interval. In this chapter, G denotes a compact group.

Definition 3.1. Let {C}) be a net in ' (Ga). One says that xA 4, 4

E -PO(G), quasi-uniformly on d if (Ox)a O ,) for each a E ( and for E > 0

and AO there exists a finite number of indices Al, . . ., An > AO such that, for each

a E G, min{II(Ox.)a - Oa I o: 1 < i < n) < e.

Definition 3.2. The unit ball in A(G) is B. For a E G, let V. denote the (closed)

span in A(G) of T7ij, I < i,j < n. We let Ba = B n , and E = U(Ba: a E d }. The A(G)-closed convex hull of E is denoted by co(E).

Proposition 3.3. B = co (E).

Proof. Let f E A(G) with Ilf IA? 1. Thus f has the Fourier series a nj wherefa = f * X E V. and zaeo na/la IIA = |If/lA [2, Chapter 7]. 0

Notation 3A. Letj denote the canonical map of A(G) into -G)* (G) A(G)**.

The unit ball in (G )* will be denoted by B**. Let r denote the weak-*

topology on ef (G )*. NowjB is contained in and T-dense in B** (see [4, p. 424D.

Proposition 3.5. coEr(jV) = B**.

Proof. It is immediate that ao'(jEt) contains jB. 0

Theorem 3.6. Let {fn) be a sequence in L<(G(). If supI I4O, I < oo and

(on)If) -n 0, f E jEr where (, -) denotes the natural pairing of ..w (G ) and

r (G )*, then O)n -4' 0 weakly in Z' (G ).




Proof. Let {4o) be norm bounded and (c1,,f) -> 0 forf E jE. Now jET is a T-compact subset of B**. It will suffice to show that (,,f ) -* 0 for f E B**.

Now letf E B**. Sincef is in the T-closed convex hull of jEr (see Proposition 3.5), there exists a probability measure It on jET which representsf (see [8, p. 5]). The result now follows from the Lebesgue dominated convergence theorem. 0

Theroem 3.7. Let ({.) C .L? (G ). If sup II 4 IL< oo and every subsequence {cj)

converges quasi-uniformly on d to 0, then 4,, -n 0 weakly in ' (Ga).

Proof. We may assume that I1i1. ? 1 and suppose )n +, 0 weakly in ?o(O ). Then there isf E jET C (Gc )* and e > 0 such that 1(4,f )I > E for infinitely many n. By reindexing, we may assume that I(pnsf )I 2 e for all n (by passing to a subsequence).

Let nl, ..., nk be such that min{II(4ni)a IL: I < i < k} < E. Now letting Ui = {g e .L'(G)*: I[(O,g)l > 4 1 < i < k, we have thatf E U = nf{U,: 1 < i < k). Thus U is a (nonempty) T-open neighborhood of f, and there is a g E A(G)suchthatg E Eandjg E U. ButE = U{Ba: a E G}andsogisin some BaO. Now I(5ni,Jg)I > E, 1 < i < k, contrary to the choice of {,n,: 1 < i < k). 0

We now prove the converse.

Theorem 3.8. Let (,,,} C Z (Ga). If {,)} -*n 0 weakly in P (Ga), then sup({114 IL) < 00 and every subsequence of {(4, converges quasi-uniformly on d to 0.

Proof. Let e > 0 and N E Z+ (positive integers) be given. Forf E B**, there is n(f ) 2 N such that 1(4n(f )Jf)I < E. Let U(f ) = (g E= B**: I(4n(f),g)I < e}, a T-open set. Now the collection of all U(f ), f E B**, covers B**; hence there is a finite subcover, U(f1), ..., U(fk). Thus for g E B**, min{l(4fl(f.),g)I: 1 < i < k) < e; and a fortiori, for a E G, min (IGn(f.))a IL: 1 < i < k) < e (pairing against B.)J 0

Remark 3.9. The above theorem is also valid for nets.

Chapter 4. T-sets. The concept of a T-set for an l.c.a. group has been useful in studying w.a.p. function on l.c.a. groups (for example see [9]). We will make the analogous definition of a T-set for the dual of a compact group.

Definition 4.1. For a, /B E G, one can consider the tensor product T. 0 To of the two representations. This tensor product decomposes into irreducible compo- nents: T, X T0 E elma,B(y)T,, where maj#(y) = .fG Xa,-X X dmG, a nonnegative integer. For E, F C G, we define E O F = {y C G: map(y) # 0,a E E,,8 c F). This operation makes ( into a hypergroup. If E 0 E C E, then E is called a subhypergroup of d. For a C G, there is a conjugate a E G such that X(x) = xa(x), for each x e G. If E C G, then E = {a: a 8 E}. We use {1) to denote the trivial representation G -3 {1}.




Proposition 4.2. For a, /3, y E G, map(y) = m,(y) = may(3)-m700) -m#, (a). Also XO, = X 2,yr majg(y)Xy, thus y m,fi (y)ny =n n1p

Definition 4.3. Let F C G. The set F is said to be a T-set if and only if (F X a) n (F ? /) is finite for all a, / E G, with a /8.

Theorem 4A. Let G = ][I,, Gn, the complete direct product of a countable collection of nontrivial compact groups {G}. Let Fn be a finite subset of G,, such that F,;\{l} # 0. Under the natural identification of Gn with a subset Of G, the set F = U1Fn is an infinite T-set.

Proof. Now 6 is the hypergroup generated by UQ cG; that is, if a E G, there exist aj E G,, 1 < < m < oo (m depending on a), such that x (x) - xa,(xl) ... X,,(xm), where x (x 1x2, .) E G (equivalently, a-a a, * ...

?am). Let a, / E G, a A. We write a=a, ? ? am and , = 81 / 3 ,?Sm,

some m E Z+. We may assume that (1) E F. Let F' = UmI. F,, and F" - (U m+iFn). If y, y' E= (lnlm+l G,)^\{l}, then a 0 y, 3 0 y' are irreducible representations of G not in Gn) , and a 0 -y # /3 0 y'. Thus F" 0 a and F" 0 /8 are disjoint sets and do not meet (H= Gn)A Further F' 0 a, F' 0 /8 are finite subsets of (IIm I Gn)^. Thus (F 0 a) n (F 0 /3) = (F' 0 a) n (F' 0 /8), a finite set. 0

Remark 4.5. The above results holds for the complete direct product of any infinite collection of compact groups. The proof is the same.

Definition 4.6. Let 4 E i2 (Ga), define the carrier of 4, cr 4, to be the set {a E G(: Oa # 0). Forf E A(G), we write crf for crf. (Observe for 4 E f (G')

that cr C c G whereas spt 4) C G.) Note that cr(fg) C crf cr g, for f, g E A(G).

Proposition 4.7. Let f E A(G) and 4 Ee c*X(G'). Then cr(f 4)) C (crf) ? (cr 4)).

Proof. Let a E G. Then a X cr(f 4) if and only if <g,f 4)> = 0 for all g E J, if and only if <fg, 0> = 0 for all g E V. if cr(fg) n cr(4) = 0 if (cr(f) 0 a) n cr(4) = 0 if and only if mag(y) = 0 for all y E cr(4), /8 E cr(f) if and only if m,y(a) = 0 for all y E cr(4), /3 E cr(f), if and only if aE-(&f)0 (cr ). O

Theorem 4.8. Let 4 e -m (G) be such that cr 4 is a T-set in (. Then 4E W(G).

Proof. We first claim that it suffices to show that U {B. 4 ): a E G) is relatively weakly (sequentially) compact; for then its norm closed convex hull, which is B * 4 D 0(4)) (Proposition 3.3) is also weakly compact by the Krein- gmulian theorem (see [4, p. 434]).




Choose a sequence {f j} c U Ba. Now if there is some ao E G' for which fn E Ba. infinitely often, then by the compactness of B,,,, * 4 has a weak cluster point. Thus we may suppose that no Ba contains more than one f. We will write An-f = f& 4' where f E B.

Suppose PIk, 82 E G such that (4i)p, # 0 and (4')#2 # 0 for infinitely many n. Now /3 e cr4, C a,, cr O only if a,, Ef cr c. Thus (?I X cr4') n ( 2 0 cr4) is an infinite set and so PI = 8(2.

Now let E > 0 and N E Z+. By the above remark lim,,(4)a = 0 for all a E G except at most one which we denote by S. By extracting a subsequence, we assume that limn(4A)a = 0 for a # 8 and limji(4)8 = a E i23(Cna). Let cr4/N

n cr4pN+l = {#3I ... *,/p} C G. Now choose n3, ..., nk 2 N such that if A # 5, II(4'n)f,I IL < E 3 < i < k, and choose no ? N such that 11(4+)8 - al; < E.

Let n1 = N and n2 = N + 1. Letting T denote the operator in -L' (G ) defined by T7 = 0, a 85, T7' = a, we have that for any a E G, min{Il I(4, - T)alI : 0 < i < k) < E. Thus {(4') converges quasi-uniformly on 6 to T. Clearly any subsequence of {4',} does the same and so by Theorem 3.7, %Pn -n T weakly. Consequently, we have that 0(4') is relatively weakly compact. 0

Chapter 5. A proper containment of eM4(d) in W(d). Recall from Definition 2.7 that at(d) is the closure of 9M(G) in ?r (a). In Theorem 2.8 we showed that .z4(d) is contained in W(d). We will, in this chapter, give examples of compact groups for which the containment is proper.

Now for G l.c.a., M(G)^- has the following characterization [2]; for 4' E CB(G), E E M(G)A if and only if {fn} C A(G), If, IIA < 1, fA -n 0 pointwise on G implies SGfn 1 dm0 -4' 0. The nonabelian analogue also holds and has an identical proof. We state the result below.

Theorem 5.1. Let G be a compact group and 4 Ee ?f (G). The following are equivalent:

(A)+ E= cA(G), (B) If a sequence {f,,) C A(G), If, IIA < 1, and fn -O" 0 pointwise on G, then

<f,o> = Tr( A 4) -__) 0.

Definition 5.2. For G a compact group and F C G, one says that F is a central Sidon set if and only if for any O E Z xE(P) (the center of ?_ (G

' F) there is a

I E E-1M(G) (the center of M(G)) such that j I F = 4. In [3], we studied central Sidon sets and showed for G an infinite compact

group that G is not a central Sidon set. It follows that ?A(G) (the center of A(G) relative to convolution) is proper in .2C(G) (the center of C(G) relative to convolution). Thus for E > 0, there is f e 2A(G) such that I1flIA = 1 and lIfILo < E. In fact, one may take f to be a central trig series; i.e. f(x) - 2-aeF Ca X(X), where F is a finite subset of G and ca E C.

Theorem 5.3. Let G = 11n Gn be the complete direct product of a countable collection of nontrivial compact groups {Gn}. Then zM-(d) is a proper closed subspace of W(A).




Proof. Since G is an infinite group, 0 is not a central Sidon set. Thus there is a central trig polynomialf1 such that Ilf1 I IA = I and I If, Ilo < 1. Let Fj = crfl.

Since cr fi is finite there is an ml E Z+ such that crf1 C ftI where HI II I Gn.

Now suppose that we have constructed central trig polynomialsfi. .f., ff and integers I < ml < m2 < ... < mk such that llflL < K/j, IlfIllA = 1, and crfj CIR, where Hj = Jj'j+ +I Gn, for j = 1, ..., k. Then there is a central trig polynomial fk+l on K = II'=mk+1 Gn such that tIfk+l IIA = 1 and IIfk+tIL < l/(k + 1) (observe that fk+l is also a trig polynomial on G with the same sup and A norms). Pick Mk+l > Mk such that crfk+l c (ll?k+1 +1 Gn)^. Thus by induction there exists a sequence of central trig polynomials {fk} on G and a sequence of finite subgroups {Hk} such that G = II' Hk, and crfk C 4k, llfkIIA =

ttfkIl < 1/k for each k Ei Z+. Now let F = U<k. Icr fk; then by Theorem 4.8, F is a T-set and so any

4) e L'S(G) with cr4 C F is in W(d). Since each fk has lIfkIIA = 1, there is 4)k E ZC'rQ(G ) such that cr4)k = crfk, II4)k IkL = 1 and <fk 4)k> = 1. Let 4' E ' (G@) be defined by 4vi = (4k). if a e crfk for some k, and q/i = 0

otherwise. Thus 4 E W(G). Furthermore, since <fk, 4> = <fk, )k> ok 0, we have 4 + 4At(G() (by Theorem 5.1). 0

Remark 5A. Observe that the above result also holds for the complete direct product of an infinite collection of nontrivial compact groups. The proof is identical with the one above. Also our proof shows that 2la(d) is proper in W(G-) n 7ZO(d).

Remark 5.5. If G is a nondiscrete l.c.a. group then the space of uniform limits on d of Fourier-Stieltjes transforms,M(G )^-,is a proper subspaceof the continuous weakly almost periodic functions on G. The general result is due to Ramirez [9].

Remark 5.6. Let Md(G) denote the space of discrete measures on G and .714d(G) its closure in -o (G'). Just as (G) has a characterization as a class of linear functionals on A(G) (Theorem 5.1), A{d(d ) may be characterized similarly. The proof is similar to the abelian case (see [2, Chapter 31). We present the following as a contrast to Theorem 5.1.

Theorem 5.7. Let G be a compact group and 0 E c' (G). The following are equivalent:

(A),oENd(G (B) If a net {fA) C A(G),I IfAIIA < 1, and fA -A 0 pointwise on G, then

<fA,4)> = Tr(4fA) -A 0. Chapter 6. A proper containment of W(d) in L' (G ). Let Z be the integers and

Z+ the positive integers. Denote by 4)+ the characteristic function of Z+. Now 4)+ E 1I(Z) bat is not w.a.p. One way to see this is to note that {R(-k)4+}k=I converges pointwise to 0 but not quasi-uniformly. This idea will show for G a complete direct product of a countably infinite collection of nontrivial compact groups that W(d) is a proper closed subspace of e' (G).




Definition 6.1. Let F C GB. We say that F is a Q-set if and only if there is a sequence {ak) C Gi such that:

(1) for a e G', (ak 0 a) n F = 0 eventually, (2) given! E Z+,thereisa E G such that U{a(a,: 1 < i < 1) C F. The sequence {ak} is called the associated sequence. Remark 6.2. (a) Let G = T, the circle group. Then F = Z+ is a Q-set in ' = Z

with associated sequence {-k})2.k. (b) Let G = JJn Z(pn), the complete direct product of a countable number of

cyclic groups Z(pn) of order p, > 1. Then F = {x E G- = 2 e.Z(pn): x = (xI, ... , xn,O, 0,0 ... ), xn # 0, n even} is a Q-set in c with associated sequence

{x(k)), x(k) (0, ... ., O, 1, O, . . . ), where 1 appears in the (2k + 1)th coordinate. (c) Let G = Fn Gn, the complete direct product of a countable set of nontrivial

compact groups. Then F- {x e G: x = al 0 an,ai E=Gi (1 < i < n), a#= (1), n even) is a Q-set in c with associated sequence {x(k)}, x(k) E G2k+l\{l})

(d) Let G =-Ap the p-adic integers. Then Gc = Z(pe), the group of unimodular complex numbers of the form x = exp(2Tril/pr+I), 0 < I < p+, 1, r E Z+. Then F = {x E Z(p"): ord x 5 p2n,n E Z+} is a Q-set in Z(p') with associated sequence Xk = exp (2Tri/p+t I).

Theorem 63. Let F be a Q-set in c with assoicated sequence {ak}. Then 4 E LO (G') defined by 40- = In. (identity operator in !13(Cn')) for a E F and

= 0 otherwise, is not in W(ci).

Proof. Let fk= XC,k/nG.. Thenfk E P. It follows from Propositions 4.2 and 4.7 that a E spt(fk * 4) implies (ak ? a) n F # 0. Thus fk * 4 k 0 pointwise on ci.

It remains to show that {fk 4 4} has no subsequence which converges to 0 quasi- uniformly. Now for 4' E lY'(0), we write ca = c.(4')IJ (a E G). Then for (3 E G, X. ( e .Z& r(G) and a simple computation shows

c.(xsO * = 1- I

n7m,(y)c7(4').

Now suppose there is I E Z+ such that for each a E c, min{II(fk * O), Ill: 1 < k < 1) < . By property 6.1(2), there is ao E G' such that {ao 0 ai: 1 < i < 11 C F. But then we will have

1(fk -Oao= Lcao(fk .)))

=1 I nm'm,Ac(cy)b n- nan, n7% E()




for mGOGk(y) # 0 only for y E ao X ak C F, cy(o) = 1 for y E F and the identity 2y ny m. (y) = na np, any a, Jl y E d (see Proposition 4.2). 0

Corollary 6.4. Let G = ll, G, be the complete direct product of a countable collection of nontrivial groups. Then W(G) is a proper closed subspace of J2 (69).

Corollary 6.5. Let G be a nondiscrete locally compact abelian group. Then there is a continuous, bounded function on 6 which is not weakly almost periodic.

Proof. If G is compact, then G is a discrete (infinite) abelian group and thus has a subgroup of the form: (a) Z, (b) Z(p*), (c) a countable infinite sum of cyclic groups (see [2, Chapter 2]). By Theorem 6.3 and Remark 6.2, our conclusion is valid provided G is: (a) T, (b) A., (c) a countably infinite product of cyclic groups. Observe that if the result is valid for A, a closed discrete subgroup of G, then it is certainly valid for G since the restriction of a continuous w.a.p. function to A is a w.a.p. function and every bounded function on A extends to a continuous bounded function on G. Thus the result holds for those G for which 6 is discrete or has an infinite discrete closed subgroup. By the principal structure theorem [10, p. 40] for l.c.a. groups we may assume G has a nonempty compact open subgroup A (otherwise G contains a euclidean space, thus Z). Since G/A is an infinite discrete group, we let f be a bounded function on d/A which is not w.a.p. Extend f to all of d by f(y + A) = f(y), y E Clearly f is not w.a.p. on G (by quasi-uniform convergence). 0

Remark 6.6. Another proof of Corollary 6.4 may be found in [1, p. 68].

III. EXTENSIONS OF THE THEORY

Chapter 7. Almost periodic functionals. One may drop the word "weakly" from Definition 2.1 and thus create a nonabelian analogue of almost periodic functions. We use the notation of Chapter 2.

Definition 7.1. For G a compact group and E E L (G), we say that 4 is almost periodic if and only if the map f H-f -) from A(G) -e ?' (G) is a compact operator. The space of all such 4) is denoted by A P(6G).

The following is clear (see Theorem 2.5 and [4, p. 486]).

Theorem 7.2. AP(d) is a closed * -submodule of W(d), and a priori of P ((G6.

T1heorem 7.3. If IL is a discrete measure in M(G) then Cu E A P(G )

Proof. If A is finitely supported then f 4 f *, is of finite rank. 0

Chapter 8. Locally compact groups. All the numbered definitions, propositions, and theorems of Chapters 2 and 7 also hold for noncompact locally compact groups, if the following replacements are made:

(i) The Fourier algebra A(G) is also defined for locally compact groups, and is a regular commutative Banach algebra of continuous functions on G vanishing at infinity (see [6]).




(ii) Replace ?/ (G') by VN(G), the weak-operator closed algebra of operators on L2(G) (left Haar measure) generated by the left translations. Each IL E M(G) is represented in VN(G) as a left convolution operator f 9- * f, f E L2(G). Again VN(G) is the dual of A(G) [61 and is an A(G)-module.

BIBLIOGRAPHY 1. R. Burckel, Weakly almost periodic functions on semigroups, Gordon and Breach, New York, 1970.

MR41 #8562. 2. C. Dunkl and D. Ramirez, Topics in harmonic analysis, Appleton-Century-Crofts, New York,

1971. 3. , Sidon sets on compact groups, Monatsh. Math. 75 (1971), 111-117. 4. N. Dunford and J. T. Schwartz, Linear operators. I: General theory, Pure and Appl. Math., vol.

7, Interscience, New York, 1958. MR 22 #8302. 5. W. Eberlein, Abstract ergodic theorems and weak almost periodic functions, Trans. Amer. Math.

Soc. 67 (1949), 217-240. MR 12, 112. 6. P. Eymard, L'algebre de Fourier d'un groupe localement compact, Bull. Soc. Math. France 92

(1964), 181-236. MR 37 #4208. 7. J. Kitchen, Normed modules and almost periodicity, Monatsh. Math. 70 (1966), 233-243. MR 33

#6297. 8. R. Phelps, Lectures on Choquet's theorem, Van Nostrand, Princeton, N.J., 1966. MR 33 # 1690. 9. D. Ramirez, Weakly almost periodic functions and Fourier-Stieltjes transforms, Proc. Amer. Math.

Soc. 19 (1968), 1087-1088. MR 38 #488. 10. W. Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Appl. Math., no. 12,

Interscience, New York, 1962. MR 27 #2808.

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF VIRGINIA, CHARLOTrESVILLE, VIRGINI 22903



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Weakly Almost Periodic Functionals on the Fourier Algebrapeople.virginia.edu/~der/pdf/der29.pdf · Stieltjes transform 9) of M(G), the measure algebra of G, into W(d). If G is compact