Weakly Almost Periodic Functionals on the Fourier Algebra Author(S): Charles F

Weakly Almost Periodic Functionals on the Fourier Algebra Author(s): Charles F. Dunkl and Donald E. Ramirez Source: Transactions of the American Mathematical Society, Vol. 185 (Nov., 1973), pp. 501-514 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/1996454 . Accessed: 29/03/2014 20:55

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This content downloaded from 128.143.23.241 on Sat, 29 Mar 2014 20:55:16 PM All use subject to JSTOR Terms and Conditions 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 transformsof 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 algebraA(G), let W(G)be the set of 4) E A(G)* (the dual of A(G)) such that the map ff-* 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 transform9) of M(G), the measure algebra of G, into W(d). If G is compact then the easily accessible structureof 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 September7, 1970. AMS (MOS) subject classifications(1970). Primary 43A10, 43-00, 43A75; Secondary 22D40, 43A07. Key wordsand phrases. Weakly almost periodic, almost periodic, means, quasi-uniformconvergence,Fourier algebra. Copyrighti 1974, AmericanMathematical Society 501

This content downloaded from 128.143.23.241 on Sat, 29 Mar 2014 20:55:16 PM All use subject to JSTOR Terms and Conditions 502 C. F. DUNKL AND D. E. RAMIREZ convergence (invented for use on scalar valued functions) to characterizeweak 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 organizationof 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)-modulepoint of view. ?11 is the main part of the paper, and deals with compact groups. Chapter 2 gives the basic definitions and propertiesof the w.a.p. functionals and the mean, and uses the ergodic semigrouptheory of Eberlein[5]. Chapter3 deals with quasi- uniform convergence,and heavily uses the theory of weak topologies on Banach spaces. In Chapter4, the T-sets are constructed.These are "thin" subsets of the dual of G, which have the propertythat any 4) E A(G)* which is carriedby one, is w.a.p. Chapters5 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)*. ?111contains Chapter7 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 preparationof this paper the authorswere partly supportedby 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 translatesof 0, denoted by 0(,O),is relativelyweakly 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 algebraunder the pointwise operationsand the norm lIf 11 = 119fllj.Now sinceA(G) _ L' (), the dualof A(G)can be identifiedwith Le(G), and the pairingis givenby = Jf (9f)(y)O(y)dmO(y),f E A(G), E Le (G), and wherem< is the Haarmeasure 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 LV(G()by = ,g E A(G).

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Fix ?J e L(G ). Let T be the bounded operator from A(G) to Le (G ) given by E Note < Now for E fH-f f *,f A(G ). If *4|L Ilf IIA41. g A(G),

= = f 9(fg)(y)4)(y)dm0(y) = 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 representationof operators with values in a commutative C*-algebra(see [4, p. 4901). The operator T induces an adjoint map r from I,3Cthe Stone-tech compactification of 6, into L(G) by the rule = Tf(y), f E A(G), y E /G3. Thus for f E A(G), y E G C f80, we have = (J * 4+)(y) = 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 weaklycompact 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. Theorem1.1. Let G be a locally compactabelian group. For 4 E8 Lj'(G() to be weaklyalmost periodic it is necessaryand sufficientthat the mapf i-+f * 4 from A(G) to CB(G) (f * 4 = J * 4) is a weakly compactoperator. Remark1.2. Choose an approximateidentity {ux} in LI(G ) (G is an l.c.a.group) such that ul= ux and jluxII, = 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. COMPACTGROUPS 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 representationsof 0. For a E G, let T, be an element of a. Then T7is a homomorphismof G into U(na), the group of na X na unitary matrices where n, is the dimension of a. We use T7(x)ijto 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,

This content downloaded from 128.143.23.241 on Sat, 29 Mar 2014 20:55:16 PM All use subject to JSTOR Terms and Conditions 504 C. F. DUNKL AND D. E. RAMIREZ let x.,(x) = trace(T.(x))= T7;(x)i,. The functionis calledthe characterof a and it is independentof the choiceof T, in a. LetX be an n-dimensionalcomplex inner product space. Let 2B(X) be the space of linear maps from X -- X. We define the operatornorm of A E .23(X)by I|AIL = sup{IAXI:t E X, |tI < 1). The traceof A, Tr A, is E (A&,) where { i}7=l is any orthonormalbasis for X and (,) denotesthe innerproduct in X. Let IAIdenote (A*A)I/2. The value IAILis the spectral radius of JAI;i.e. max{Xi:I ? i < n) whereXi are the eigenvaluesof JAI. Let p be a set {4-Oaa e G'where . E 2B(Cna))such that sup{jlk.IL: a E G} < oo. The set of all such Ois denotedby eP (G). It is a Banachalgebra under the normIl4IO1 = sup{lka IL: a E G) and coordinatewiseoperations. Let M(G) denote the measurealgebra of G, that is, the space of finite regularBorel measureson G with convolutionas the multiplication.For , e M(G), the - Fourier-Stieltjestransform of t, js is a matrix-valuedfunction defined for a E by a Gi = f 7;(x-l)dA(x). Note that ,i E Z'(). We sometimes write 9,L for ii. Let A E B(X) whereX is a finite-dimensionalcomplex inner product space. We define the dual norm to 11-IIby IfAII,= sup{ITr(AB)I:IIBIO < 1}. This norm can be also characterizedby IAII, = Tr(|AJ).For 0 E t?O(G), we put II411,= jm =, n II II1. Let -l '(G')= {, E L` (G'): 1I11< 001. The space XL(G) is a Banach space under the norm 11111.For ECe X(d), let Tr(4) = xE na Tr(O.). We will now defineA(G), the Fourieralgbra 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 - Ea jIn. < oo. Note that A(G) is isomorphicto -el(G') by f -+ 9f because for any 4) E /P(I) the functionf(x) = Eaeo na Tr(4)hT,(x)) is in A(G); further,

lIfL= sup{J naTr(4 Ta(X))f.x)|X =

We now recall the followingfacts (see [2, Chapter8]): A(G) -' () by f -* 9fjf E A(G); A(G)is an algebraunder pointwise multiplication; the dual of -I (G) is 12 (G) and the correspondenceis given by 4Tr)(>), ) E ?' (O), O E (O ). Thus the dual of A(G)can be identifiedwith -'(G(i) and the pairingis given by = Tr(oJ), f E A(G), 4 E (G). The module action of ?c1(G) over A(G) is definedby the following:f E A(G), 4 E ?0(d), f *0 E PO(d) by = , g E A(G). Note lIf-,OI < IIflIA1II1IL Definition2.1. For G a compactgroup and 4 e 1t(GQi), we say that 4 is weaklyalmost periodic (w.a.p.) if and only if the mapf t f - 4 fromA(G) to -LI(Gc)is a weaklycompact operator. The space of all such 4 is denotedby w(d).

This content downloaded from 128.143.23.241 on Sat, 29 Mar 2014 20:55:16 PM All use subject to JSTOR Terms and Conditions WEAKLYALMOST PERIODIC FUNCTIONALS 505 We denotethe unit ball in A(G)by B. We denoteby P the set of continuous positivedefinite functions on G whichat e, the identityof 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. Proposition2.2. For 4) E ?' (c), 4 E=W(G ) if and only if B .) = {f *0: f E B} is relativelyweakly compact in ?4 (G ) if and only if P*) = {(f 4): f E P} is relativelyweakly compact in eL'(G ). Proof.We needonly observethat any hermitianoperator is a differenceof two positivedefinite operators. Thus B is containedin the balancedconvex hull of 2P. O - Definition2.3. For 4 E (G ), we define0(4)) to be the set P 4)O.(Observe that 0(4)) is convex.) Remark2A. For 4 E eP (O), to show that 0(4) is relativelyweakly compact it is necessaryand sufficientto show that 0(4)) is relativelyweakly sequentially compact;equivalently, 0(4) is relativelyweakly countably compact (see [4, p. 430]).

Theorem 2.5. W(O) is a closed *-subspaceof ?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 operatornorm of T, 11TII, is the sameas 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 uniformlimit of weaklycompact operators is weaklycompact. The *-invariancefollows since 0(4*) = 0(4))* and 4) 0 * is a weaklycontinuous map. a Theorem2.6. W(6) is a moduleover A(G). Proof.Let 4 EGW(G) andf E A(G). We need to show thatf 4 E W(G). We observethat B - (f 4)) C (jlfjlAB) *4. 0 Definition2.7. Let at(d) be the closureof 9M(G) in 2*(Gc). Theorem2.8. The algebra .Al(O) is containedin W(O). Proof. It will suffice to consider a positive measure ,u E M(G) with 11l41=I. Let =# be the Hilbertspace L2(G,j). Considerthe map S: IC (G) by (g,Sf) = f (fg)^d,u, g E A(G), f E W. Now f(gSf )I < IIg9ILYlifIL < 11gILA lifL1,, and thusS is stronglycontinuous and henceweakly continuous [4, p. 422]. Let {fnQC A(G), IIfnIIA< 1. Then 4fn)is containedin the unit ball of 4# which is weaklysequentially compact. Thus thereis a subsequencefn -k h weaklyin cJf.Thus Sfn ---kSh weakly. It remains to observe that Sf = f* ji, f E A(G). This followsfrom:

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

(=, f,gEA(G). 0

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Note that this implies that Co(d) C W(d), where e80(d)-( E ?~"(G): II4a)It> 8 for only finitely many a E Gdfor 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 semigroupof operators on e' (d). Definition 2.9. The semigroupP 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) pAis a linear transformationof .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'). Theorem2.10. The semigroupP is ergodic.

Proof. Let (A}be a neighborhood basis for e E G. Let UAE P with support UAC 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 transformationPA from W(d) to W(d) by pA4)= UA* 4, 4 E W(G). Properties2.9(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 thereforeenough 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 Theorem2.11 To every 4 E W(G) therecorresponds a constantdenoted by 0(4) such thatfor 4, 4, E W(G) one has (1) 0(a4 + 4,)= a0(4)) + 0(4,)and I0(4)I < II 1 (a E C), (2) 0(I ) = 1 (I denotesthe identity operator in ZL0(G)), (3) if 4)a> Oforall a, thenf(4)) 2 0, (4) UA4* -*A 0(4))I in the normof 2 (G'), {UAIas 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) uniquelydetermine 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 operatorI 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

This content downloaded from 128.143.23.241 on Sat, 29 Mar 2014 20:55:16 PM All use subject to JSTOR Terms and Conditions WEAKLYALMOST PERIODIC FUNCTIONALS 507 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 = 9(Jd,).u 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 AOthere exists a finite number of indices Al, . . ., An > AOsuch 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) ofT7ij, 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/laIIA = |If/lA[2, Chapter7]. 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 supII4O, 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 ).

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

Theroem3.7. Let ({.) C .L? (G ). If sup II4 IL< oo and everysubsequence {cj) ' convergesquasi-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)aIL: 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.

Theorem3.8. Let (,,,} C Z (Ga).If {,)} -*n 0 weakly in P (Ga), then sup({114IL) < 00 and everysubsequence of {(4, convergesquasi-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 Remark3.9. The above theorem is also valid for nets.

Chapter4. 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 Toof the two representations.This tensor product decomposes into irreduciblecompo- nents: T, XT0 Eelma,B(y)T,, where maj#(y)= .fG Xa,-XX 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}.

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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 collectionof nontrivialcompact groups {G}. Let Fnbe a finite subset of G,,such that F,;\{l} # 0. Under the natural identificationof Gnwith a subset Of G, the set F = U1Fn is an infinite T-set.

Proof. Now 6 is the hypergroupgenerated by UQcG; 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 (x1x2, .) E G (equivalently, a-a a, * ?am). Let a,/ E G, a A. We write a=a, ? ? am and , /= 3 81,?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 # /30 y'. Thus F" 0 a and F" 0 /8are disjoint sets and do not meet (H= Gn)A FurtherF' 0 a, F' 0 /8are 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 Cc 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 = 0 for all g E J, if and only if = 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), /3E 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]).

This content downloaded from 128.143.23.241 on Sat, 29 Mar 2014 20:55:16 PM All use subject to JSTOR Terms and Conditions 510 C. F. DUNKLAND D. E. RAMIREZ Choose a sequence{f j} c U Ba. Now if there is some ao E G' for which fnE Ba.infinitely often, then by the compactnessof B,,,, *4 has a weakcluster point.Thus we may supposethat no Ba containsmore than one f. We will write An-f= f& 4'where f E B. SupposePIk, 82 E G suchthat (4i)p, # 0 and (4')#2 # 0 for infinitelymany n. Now /3 e cr4, C a,, cr O only if a,, Ef crc. Thus (?I X cr4') n ( 2 0 cr4) is an infiniteset and so PI = 8(2. Now let E > 0 and N E Z+. By the aboveremark lim,,(4)a = 0 for all a E G except at most one which we denote by S. By extractinga 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,IIL < E 3 < i < k, and choose no ? N suchthat 11(4+)8- al; < E. Let n1 = N and n2 = N + 1. LettingT denotethe operatorin -L' (G) definedby T7= 0, a 85,T7' = a, we have that for any a E G, min{IlI(4, - T)alI : 0 < i < k) < E. Thus {(4') converges quasi-uniformlyon 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 relativelyweakly compact. 0 Chapter5. A propercontainment of eM4(d)in W(d). Recall from Definition 2.7 that at(d) is the closureof 9M(G) in ?r (a). In Theorem2.8 we showedthat .z4(d) is contained in W(d). We will, in this chapter, give examples of compact groupsfor whichthe containmentis proper. Now for G l.c.a., M(G)^- has the following characterization[2]; for 4' E CB(G), E M(G)A if and only if {fn} C A(G), If,IIA < 1, fA -n 0 pointwise on G impliesSGfn 1 dm0 -4' 0. The nonabeliananalogue also holdsand has an identicalproof. We statethe resultbelow. Theorem5.1. Let G be a compactgroup and 4 Ee?f (G). The following are equivalent: (A)+ E=cA(G), (B) If a sequence{f,,) C A(G), If,IIA < 1, andfn -O"0 pointwiseon G, then A = Tr( 4) -__)0. Definition 5.2. For G a compact group and F C G, one says that' F is a central Sidonset if and only if for any O E Z xE(P)(the centerof ?_ (G F) thereis a

I E-1M(G)E (the centerof M(G)) such that j I F = 4. In [3], we studiedcentral Sidon sets and showedfor G an infinitecompact groupthat G is not a centralSidon set. It followsthat ?A(G)(the centerof A(G) relativeto convolution)is properin .2C(G) (the center of C(G) relativeto 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), whereF is a finitesubset of G and ca E C. Theorem5.3. Let G = 11nGn be the complete direct product of a countable collectionof nontrivialcompact groups {Gn}. Then zM-(d)is a properclosed subspace of W(A).

This content downloaded from 128.143.23.241 on Sat, 29 Mar 2014 20:55:16 PM All use subject to JSTOR Terms and Conditions WEAKLYALMOST PERIODIC FUNCTIONALS 511 Proof.Since G is an infinitegroup, 0 is not a centralSidon set. Thus thereis a centraltrig polynomialf1 such that Ilf1IIA = I and IIf, 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 supposethat we haveconstructed central trig polynomialsfi. .f., ff and integersI < ml < m2< ... < mk such that llflL Mksuch that crfk+l c (ll?k+1 +1 Gn)^.Thus by induction thereexists a sequenceof centraltrig polynomials {fk} on G and a sequenceof 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

4)k E ZC'rQ(G) such that cr4)k = crfk, II4)kIkL = 1 and = 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 = ok 0, we have 4 + 4At(G()(by Theorem5.1). 0 Remark5A. Observethat the above resultalso holds for the completedirect productof an infinitecollection of nontrivialcompact groups.The proof is identicalwith the one above. Also our proof shows that 2la(d) is properin W(G-) n 7ZO(d). Remark5.5. If G is a nondiscretel.c.a. group then the spaceof uniformlimits on d of Fourier-Stieltjestransforms,M(G )^-,is a propersubspaceof the continuous weaklyalmost periodic functions on G. The generalresult is due to Ramirez[9]. Remark5.6. Let Md(G)denote the space of discretemeasures on G and .714d(G)its closurein -o (G').Just as (G) has a characterizationas a class of linearfunctionals on A(G)(Theorem 5.1), A{d(d) maybe characterizedsimilarly. The proof is similarto the abeliancase (see [2, Chapter31). We presentthe followingas a contrastto Theorem5.1. Theorem5.7. Let G be a compactgroup and 0 E c'(G). Thefollowing are equivalent: (A),oENd(G (B) If a net {fA)C A(G),I IfAIIA< 1, and fA -A 0 pointwise on G, then = Tr(4fA) -A 0. Chapter6. A propercontainment of W(d)in L' (G). Let Z be the integersand Z+ the positiveintegers. Denote by 4)+the characteristicfunction 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 convergespointwise to 0 but not quasi-uniformly.This idea will show for G a completedirect product of a countablyinfinite collection of nontrivialcompact groupsthat W(d)is a properclosed subspace of e' (G).

This content downloaded from 128.143.23.241 on Sat, 29 Mar 2014 20:55:16 PM All use subject to JSTOR Terms and Conditions 512 C. F. DUNKL AND D. E. RAMIREZ

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 Gisuch 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. ' Remark6.2. (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 = JJnZ(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 numbersof 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 44} has no subsequencewhich 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

I c.(xsO* =1- 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, E()n7%

This content downloaded from 128.143.23.241 on Sat, 29 Mar 2014 20:55:16 PM All use subject to JSTOR Terms and Conditions WEAKLYALMOST PERIODIC FUNCTIONALS 513 for mGOGk(y)# 0 only for y E ao X ak C F, cy(o) = 1 for y E F and the identity 2y nym. (y) = nanp, any a, Jl y E d (see Proposition 4.2). 0 Corollary6.4. Let G = ll, G, be the complete direct product of a countable collectionof nontrivialgroups. Then W(G)is a properclosed subspaceof J2 (69). Corollary6.5. Let G be a nondiscretelocally compactabelian group. Then there is a continuous,bounded function on 6 whichis not weaklyalmost 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 subgroupof G, then it is certainlyvalid for G since the restrictionof 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 structuretheorem [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 Clearlyf is not w.a.p. on G (by quasi-uniformconvergence). 0 Remark6.6. Another proof of Corollary6.4 may be found in [1, p. 68]. III. EXTENSIONSOF THE THEORY Chapter7. Almost periodicfunctionals. 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. Definition7.1. For G a compact group and 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]). Theorem7.2. AP(d) is a closed *-submodule of W(d), and a priori of P ((G6.

T1heorem7.3. If IL is a discretemeasure in M(G) then CuE A P(G ) Proof.If Ais finitelysupported then f 4 f *, is of finiterank. 0 Chapter8. Locallycompact groups. All the numbered definitions, propositions, and theorems of Chapters 2 and 7 also hold for noncompact locally compact groups, if the following replacementsare made: (i) The Fourier algebraA(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]).

This content downloaded from 128.143.23.241 on Sat, 29 Mar 2014 20:55:16 PM All use subject to JSTOR Terms and Conditions 514 C. F. DUNKL AND D. E. RAMIREZ (ii) Replace ?/ (G') by VN(G), the weak-operatorclosed algebra of operators on L2(G) (left Haar measure)generated by the left translations.Each ILE 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) [61and is an A(G)-module.

BIBLIOGRAPHY 1. R. Burckel, Weaklyalmost periodic functions on semigroups,Gordon and Breach,New York, 1970. MR41 #8562. 2. C. Dunkl and D. Ramirez, Topics in harmonicanalysis, Appleton-Century-Crofts,New York, 1971. 3. , Sidon sets on compactgroups, Monatsh. Math. 75 (1971), 111-117. 4. N. Dunford and J. T. Schwartz,Linear operators.I: Generaltheory, Pure and Appl. Math., vol. 7, Interscience,New York, 1958. MR 22 #8302. 5. W. Eberlein, Abstractergodic theorems and weak almostperiodic functions, Trans. Amer. Math. Soc. 67 (1949), 217-240. MR 12, 112. 6. P. Eymard, L'algebrede Fourier d'un groupe localementcompact, Bull. Soc. Math. France 92 (1964), 181-236. MR 37 #4208. 7. J. Kitchen, Normedmodules and almostperiodicity, Monatsh. Math. 70 (1966), 233-243. MR 33 #6297. 8. R. Phelps, Lectureson Choquet'stheorem, Van Nostrand, Princeton, N.J., 1966. MR 33 # 1690. 9. D. Ramirez, Weaklyalmost periodic functions and Fourier-Stieltjestransforms, 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.

DEPARTMENTOF MATHEMATICS,UNIVERSITY OF VIRGINIA,CHARLOTrESVILLE, VIRGINI 22903

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