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Let G G be a locally compact group and let A ( G ) A(G) be its Fourier algebra. A closed subset H H of G G is said to be a Helson set if the restriction homomorphism ϕ : A ( G ) → C 0 ( H ) \phi :A(G)\rightarrow C_{0}(H) , ϕ ( a ) = a | H \phi (a)=a_{|H} , is surjective. In this paper, under the hypothesis that G G is amenable, we prove that every Helson subset H H of G G that is also a set of synthesis is a Ditkin set. This result is new even for G = R G=\mathbb {R} .

In this paper, we consider various extension problems associated with elements in the closure with respect to either the multiplier norm or the completely bounded multiplier norm of the Fourier algebra of a locally compact group. In particular, we show that it is not always possible to extend an element in the closure with respect to the multiplier norm of the Fourier algebra of the free group on two generators to a multiplier of the Fourier algebra of S L ( 2 , R ) SL(2,\mathbb {R}) .

We observe that for a large class of non-amenable groups $G$, one can find bounded representations of $A(G)$ on Hilbert space which are not completely bounded. We also consider restriction algebras obtained from $A(G)$, equipped with the natural operator space structure, and ask whether such algebras can be completely isomorphic to operator algebras; partial results are obtained, using a modified notion of Helson set which takes account of operator space structure. In particular, we show that if $G$ is virtually abelian, then the restriction algebra $A_G(E)$ is completely isomorphic to an operator algebra if and only if $E$ is finite., v3: 10 pages, minor edits and slight change to title from v2. Final version, to appear in Proc. Amer. Math. Soc

The Radon-Nikodym property for the Banach algebras A p r ( G ) = A p ∩ L r ( G ) A_p^r(G)=A_p\cap L^r(G) , where A 2 ( G ) A_2(G) is the Fourier algebra, is investigated. A complete solution is given for amenable groups G G if 1 > p > ∞ 1>p>\infty and for arbitrary G G if p = 2 p=2 and A 2 ( G ) A_2(G) has a multiplier bounded approximate identity. The results are new even for G = R n G=\mathbb {R}^n .

Let A ( G ) A(G) be the Fourier algebra of a locally compact group and U C B ( G ^ ) UCB(\hat {G}) the C ∗ C^* -algebra of uniformly continuous linear functionals on A ( G ) A(G) . We study how the centre problem for the algebra U C B ( G ^ ) ∗ UCB(\hat {G})^* (resp. A ( G ) ∗ ∗ A(G)^{**} ) is related to the centre problem for the algebras U C B ( H ^ ) ∗ UCB(\hat {H})^* (resp. A ( H ) ∗ ∗ A(H)^{**} ) of σ \sigma -compact open subgroups H H of G G . We extend some results of Lau-Losert on the centres of U C B ( G ^ ) ∗ UCB(\hat {G})^* and A ( G ) ∗ ∗ A(G)^{**} .

For a locally compact group $G$, let $A(G)$ denote its Fourier algebra and $\hat{G}$ its dual object, i.e. the collection of equivalence classes of unitary represenations of $G$. We show that the amenability constant of $A(G)$ is less than or equal to $\sup \{\deg(\pi) : \pi \in \hat{G} \}$ and that it is equal to one if and only if $G$ is abelian., Comment: LaTeX2e; 11 pages; some more minor revisions

In this paper we show that if G G is an amenable locally compact group and if H H is a closed subgroup, then the ideal I ( H ) I(H) has an approximate identity of norm 2. 2. If H H is not open, this bound is the best possible.

For a locally compact group G and 1 0, there exists a compact subset K of G such that | | 0, there exists a compact subset K of G such that |u, f| 0 and any compact subset K of G, there exists some f ∈ X with ∥f∥ | > ∥u∥ - ∈. Some results of Flory (1971) and Miao (1999) can be obtained from our main theorems by taking p = 2 and X as some C*-subalgebras of PMp(G).

The purpose of this article is to prove the following result. Let G be a locally compact group, A(G) the Fourier algebra of G, and S(G) = {u ∈ A(G):? c > 0 such that ∥ uv ∥ A(G) ≤ c ∥ v ∥∞ V v E A(G)}. Then G is a discrete group ⇔ S(G) ¬= {0}.

In this paper, we study the ideals in the Fourier algebra of a locally compact group G G which are complemented by an invariant projection. In particular we show that when G G is discrete, every ideal which is complemented by a completely bounded projection must be invariantly complemented. Perhaps surprisingly, this result does not depend of the amenability of the group or the algebra, but instead relies on the operator biprojectivity of the Fourier algebra for a discrete group.

We show that for any locally compact group G, the Fourier algebra A(G) is operator weakly amenable.

Let G be a locally compact group, A(G) the Fourier algebra of G and VN(G) the von Neumann algebra generated by the left regular representation of G. We introduce the notion of X-spectral set and X-Ditkin set when X is an A(G)-invariant linear subspace of VN(G), thus providing a unified approach to both spectral and Ditkin sets and their local variants. Among other things, we prove results on unions of X-spectral sets and X-Ditkin sets, and an injection theorem for X-spectral sets.

Keywords: convolution operator ; pseudomeasure ; amenable group ; spectral synthesis ; Ditkin set ; Fourier algebra ; Figa-Talamanca Herz algebra Reference CAHRU-ARTICLE-2001-001doi:10.1090/S0002-9939-00-05874-3 Record created on 2008-12-10, modified on 2016-08-08

Let G G be a locally compact group. Among other things, we proved in this paper that for an IN-group G G , the extreme points of the unit ball of the Fourier-Stieltjes algebra B ( G ) B(G) are not in the Fourier algebra A ( G ) A(G) if and only if G G is non-compact, or equivalently, there is no irreducible representation of G G which is quasi-equivalent to a subrepresentation of the left regular representation of G G if and only if G G is non-compact. This result is a non-commutative version of the following well known result: For any locally compact group G ^ \widehat G , the extreme points of the unit ball of the measure algebra M ( G ^ ) M(\widehat G) are not in the group algebra L 1 ( G ^ ) L^{1}(\widehat G) if and only if G ^ \widehat G is non-discrete. On the other hand, we also showed that if B ( G ) B(G) has the RNP, then there are extreme points of the unit ball of B ( G ) B(G) that are in A ( G ) A(G) . Since it is well known there are non-compact locally compact group G G for which B ( G ) B(G) has the RNP, there exist non-compact locally compact groups G G where extreme points of the unit ball of B ( G ) B(G) can be in A ( G ) A(G) . This shows that the condition G G be an IN-group cannot be entirely removed.

We present a new proof for Morita equivalence of twisted crossed products by coactions within the abstract context of crossed products of Hilbert C∗-modules. In this context we are free from representing all C∗-algebras and Hilbert C∗-modules on Hilbert spaces. The notion of Morita equivalence of twisted coactions was introduced in [B]. In [B, Theorem 3.3] we established conditions on twisted coactions which are sufficient to ensure Morita equivalence of the corresponding crossed product C∗-algebras. Later [ER] gave a shorter proof for this result using their results on multipliers of imprimitivity bimodules. However in the proofs of both [B] and [ER], all C∗algebras and Hilbert C∗-modules need to be represented on Hilbert spaces. In this paper we present a new proof for [B, Theorem 3.3] based on the notion of crossed products of Hilbert C∗-modules introduced in [B2]. Crossed products of Hilbert C∗-modules in [B2] were defined as subspaces of adjointable operators between Hilbert C∗-modules. In this abstract context, we are free from representing all C∗-algebras and Hilbert C∗-modules on Hilbert spaces as in [B] and [ER]. As a consequence, the proof here is shorter and more elegant than that of [B]. Our approach is close to the spirit of [BS], and different from [ER]. Throughout this paper G is a locally compact group and N is a closed normal amenable subgroup of G. Recall from [M, Lemma 3] that there is a surjective homomorphism Ψ from C∗ r (G) into C ∗ r (G/N) such that Ψ(λ (r)) = λ (qN (r)), where q N : G → G/N is the quotient map, λ and λ are the left regular representations ofG andG/N . We denote by WG the unitary operator on L 2(G×G) defined by [WGξ](r, s) = ξ(r, r −1s). If f is an element of the Fourier algebra A(G), then Sf (WG) = Mf . Here Sf denotes the slice map, see [LPRS, §1]. To apply [B2, Theorem 1.6] to this paper, we need to show that WG is a regular multiplicative unitary. For any ξ, η ∈ L(G), we define ωη,ξ = 〈Tξ|η〉, ∀T ∈ B(L(G)). Then for any ω = ωη,ξ, we have 〈(id⊗ ω)(WG)ξ′|η′〉 = 〈Mω◦λGξ′|η′〉, ∀ξ′, η′ ∈ L(G). It then follows that ŜWG = C0(G), and the crossed product of [B2, Proposition 1.5] is just the crossed product of [LPRS, Definition 2.4]. The unitary operator Received by the editors October 23, 1995 and, in revised form, February 6, 1996. 1991 Mathematics Subject Classification. Primary 46L05, 22D25. c ©1997 American Mathematical Society 2109 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

We show that every scalar valued continuous function f ( g ) f(g) on a compact group G G may be written as f ( g ) = ⟨ Φ ( g ) ξ , η ⟩ f(g) = \langle \Phi (g) \xi , \eta \rangle for all g ∈ G g \in G , where ξ , η \xi , \eta are vectors in a separable Hilbert space H \mathcal {H} , and Φ ( g ) \Phi (g) is a strongly continuous unitary valued function on G G which is a product of unitary representations and antirepresentations of G G on H \mathcal {H} . This product is countable, but always converges uniformly on G G . Moreover the supremum norm of f f is matched by ‖ ξ ‖ ‖ η ‖ \Vert \xi \Vert \Vert \eta \Vert . This may be viewed as a ‘Fourier product representation’ for f f , and complements a result of Eymard for the Fourier algebra. For ‘Fourier polynomials’ we show that the Hilbert space \mathcal {} may be taken to be finite dimensional, and the product finite, which is more or less obvious except in that we are able to match the correct norm. The main ingredients of the proof are the Peter-Weyl theory, Tannaka’s duality theorem, and a method developed with Paulsen using a characterization of operator algebras due to the author, Ruan and Sinclair. We also give the analogues of these formulae for compact quantum groups.

Let G be a locally compact group. We will consider amenability and weak amenability for Banach algebras which are quotients of the second dual of the Fourier algebra. In particular, we will show that if A(G)** is weakly amenable, then G has no infinite abelian subgroup.

Let G G be a locally compact group, and let G d G_{d} denote the same group G G with the discrete topology. There are various C ∗ -algebras C^{*}\text {-algebras} associated to G G and G d . G_{d}. We are concerned with the question of when these C ∗ -algebras C^{*}\text {-algebras} are isomorphic. This is intimately related to amenability. The results can be reformulated in terms of Fourier and Fourier-Stieltjes algebras and of weak containment properties of unitary representations.

In this paper, it is shown that for every discrete group G and 1 > p > ∞ , A p ( G ) ⋅ P M p ( G ) 1 > p > \infty ,{A_p}(G) \cdot P{M_p}(G) is closed in P M p ( G ) P{M_p}(G) if and only if G is amenable.

It is shown that the completion of A ( G ) A(G) in either the multiplier norm or the completely bounded multiplier norm is a Banach algebra without discontinuous derivations when G G is either F 2 {F_2} or SL ⁡ ( 2 , R ) \operatorname {SL}(2,\mathbb {R}) .

Let G be a locally compact group. We show that any one of the spaces U C B ( G ^ ) , W A P ( G ^ ) , A P ( G ^ ) UCB(\hat G),WAP(\hat G),AP(\hat G) , and C δ ∗ ( G ) C_\delta ^ \ast (G) is Asplund if and only if the group G is finite. We also show that any one of the spaces V N ( G ) , U C B ( G ^ ) VN(G),UCB(\hat G) , and C δ ∗ ( G ) C_\delta ^\ast (G) has the DPP if and only if the group G has an abelian subgroup of finite index.

Let A ( G ) A(G) denote the Fourier algebra of the locally compact group G G . We show that for a large class of groups an ideal I I in A ( G ) A(G) has a Δ \Delta -weak bounded approximate identity if and only if it has a bounded approximate identity.

Local techniques introduced by Saeki and Stegeman are employed to give conditions for unions and finite intersections of S-sets to be S-sets. Using local techniques we study unions and intersections of sets of spectral synthesis (S-sets). Beyond the well-known fact that the intersection of two S-sets need not be an S-set, there appears to be no known result about intersec- tions. It is not known whether the union of two S-sets is an S-set, but there are some partial results (e.g., Saeki (2) and Warner (6)). We give simple sufficient conditions for finite intersections and (possibly infinite) unions of S-sets to be S-sets. We generally follow the notation and terminology of Stegeman (4). A(G) denotes the Fourier algebra of a locally compact abelian group G. Suppose

Denote by B2(G) the Herz-Schur multiplier algebra of a locally compact group G and by B2x(G) the closure of the Fourier algebra in the topology of pointwise convergence boundedly in the norm of B2(G). G is said to be weakly amenable if B2X(G) = B2(G). We show that every amal- gamated product of a countable collection of locally compact amenable groups over a compact open subgroup is weakly amenable. This improves and extends previous results that hold for amalgams of compact groups. Let G be a locally compact group, A(G) its Fourier algebra and B(G) its Fourier-Stieltjes algebra. Denote by Bx(G) the closure of A(G) in the topol- ogy of uniform convergence on compact sets, boundedly in norm. Then G is amenable if and only if B^(G) = B(G). Moreover, if G is amenable, then B(G)=JtA{G) (the algebra of multipliers of A(G)). Recent papers on amenability have devoted a significant amount of attention to the Herz-Schur multiplier algebra B2(G) (He). It was noted in (BF) that B2(G) coincides with the algebra Jf0A(G) of completely bounded multipliers of A(G), introduced in (dCH), where many of its interesting properties were investigated. Amenability can be characterized in terms of B2(G) as follows. G is amenable if and only if B(G) = B2(G) ((Lo); for discrete groups, (Bo)). Ac- tually, for the purpose of studying amenability, the multiplier algebra B2(G) is better suited than J?A , because it has nicer functorial properties. For instance, the Herz-Schur multiplier constant AG (i.e., the infimum of the norms of all approximate identities in B2(G)) is a von Neumann algebra invariant ((Ha2); see (CoH) for a deep study of this invariant on simple Lie groups of real rank 1), whereas a similar statement does not hold for J(A. Above all, B2(G) is nicer in taking products. In fact, B2(GX)®B2(G2) c B2(GX x G2) (dCH), but the same statement does not hold for JfA . This paper makes use of B2(G) to study a property related to amalgamation for another type of products: free products with amalgamation. This property, studied in §2, can be naturally phrased in

A condition is given on the invertible elements of a commutative Banach algebra B which implies that the radical of B is not complemented by any closed subalgebra. This condition is then applied to show that certain quotient algebras of group algebras have an uncomplemented radical. I. Let B be a commutative complex Banach algebra with identity and let S be a closed ideal of B. We say that S is algebraically complemented in B (by C) if there exists a subalgebra C of B such that B = C © S. If a closed C can be found, we say that S is strongly complemented in B. An invertible element x G B is said to be doubly power bounded (DPB) if the doubly infinite sequence {||xn|| : n G Z} is bounded. We denote the radical of B by R. Our main result is as follows: THEOREM 1. Suppose the DPB elements of B span a dense subspace and that S is a nonzero closed ideal of B contained in R. Then S is not strongly complemented in B. Theorem 1 is applied to quotient algebras of group algebras to obtain a large class of Banach algebras with uncomplemented radical, as follows. Let Q be a nondiscrete locally compact abelian group, and let A.{Q) denote the Fourier algebra of Q (see e.g. [15]). For £ a compact subset of Q, let I() and 3{£) denote, respectively, the largest and smallest closed ideals of A(£) with hull equal to £. If B = &{Q)/3{£), then R = I{£)/3{£). By definition, £ is not of synthesis when R ^ (0). We have THEOREM 2. If £ is not of synthesis, then any nonzero closed ideal of B = A.{§) / 3{£) contained in R = l{£)/3{£) is not strongly complemented in B. A compact subset £ of Q is said to be a Helson set if the restriction algebra, A.{Q)1{£) is all of C(£), the continuous functions on £. By a theorem of Bade and Curtis [4, Theorem 4.1] we have COROLLARY 1. If £ is a Helson set not of synthesis, then the radical of B = A{C)/3{£) is not algebraically complemented in B. Helson sets not of synthesis exist in any Q [19]. The above corollary was conjectured in [1]. In §11 we give the proofs of the above theorems. In §111 we make some remarks which include further results and connections between our results and the existing literature. Received by the editors March 5, 1986. The results in this paper were presented to the Winter Meeting of the A.M.S., January 7, 1986. They are a portion of the results presented to the International Congress of Mathematicians, August 5, 1986. 1980 Mathematics Subject Classification. Primary 43A20, 46J35; Secondary 43A45, 43A46. Second author supported in part by N.S.F. Grant DMS-8320479. ©1987 American Mathematical Society 0002-9939/87 SI.00 + $.25 per page

The Banach algebra A A is said to have the bounded power property if for any x ∈ A x \in A , with | | x | | s p = lim n → ∞ | | x n | | 1 / n ⩽ 1 ||x|{|_{sp}} = {\lim _{n \to \infty }}||{x^n}|{|^{1/n}} \leqslant 1 , one has sup n | | x n | | > ∞ {\sup _n}||{x^n}|| > \infty . It has been shown by B. M. Schreiber [9, Theorem (8.6)] that, if G G is a locally compact abelian group, then the Fourier algebra A ( G ) = L 1 ( Γ ) ∧ A(G) = {L^1}{(\Gamma )^ \wedge } has the bounded power property, if and only if G G is discrete. We improve this result in the Theorem. Let G G be an arbitrary locally compact group and 1 > p > ∞ 1 > p > \infty . Then A p ( G ) {A_p}(G) has the bounded power property if and only if G G is discrete. Our proof, even for abelian G G and p = 2 p = 2 (then A 2 ( G ) = A ( G ) {A_2}(G) = A(G) is the usual Fourier algebra of G G ), is much simpler and entirely different from that of [9].

In this paper it is shown that every nondiscrete LCA group contains a compact independent Helson set which disobeys spectral synthesis. In [3] T. W. Korner has constructed an independent compact H 1-set of type M. This result solves negatively the long-standing problem whether or not every Helson set obeys spectral synthesis. K6rner's construction of such a set is, however, very complicated, and R. Kaufman [2] has simplified it (see also [6]. In this paper, we modify Kaufman's method to prove that every nondiscrete metrizable LCA group contains an independent compact Helson set of type M. Consequently it is shown that every nondiscrete LCA group contains a Helson set which disobeys spectral synthesis. Let G be a LCA group with dual G. We denote by A(G) and PM(G) the Fourier algebra on G and the conjugate space of A(G), respectively. Each element of PM(G) is called a pseudo-measure on G. For / e A(G) and P e PM(G), we define (/, p) =( * Pi)(1)= f X (x)P(X )dx, where f and P denote the functions in L (G) and in L(G) whose (inverse) Fourier transforms are / and P, respectively. If t e M(G) and -(x) = ftX(x) dji(X) (x e G), then we denote by jP the pseudo-measure on G defined by the requirement (WP)= ,u * P. It is well known that if P e PM(G) has compact support, then P can be chosen from the space C(G); we will always do this. For such a P, (-, P) = (f * P)(1) (G e M(G)) is well defined, and we have (X, yP) = P(Xly 1j (XX y e G). A pseudo-function on G is any pseudo-measure whose Fourier transform is a continuous function on G which vanishes at infinity. The space of all pseudo-functions is denoted by PF(G). Received by the editors July 25, 1973. AMS (MOS) subject classifications (1970). Primary 43A45; Secondary 43A20.

Let G G be a locally compact group. We prove in this paper that G G is amenable if and only if the group algebra L 1 ( G ) {L_1}\left ( G \right ) (respectively the measure algebra M ( G ) M\left ( G \right ) ) satisfies a finite-dimensional invariant subspace property T ( n ) T\left ( n \right ) for n n -dimensional subspaces contained in a subset X X of a separated locally convex space E E when L 1 ( G ) {L_1}\left ( G \right ) (respectively M ( G ) M\left ( G \right ) ) is represented as continuous linear operators on E E . We also prove that for any locally compact group, the Fourier algebra A ( G ) A\left ( G \right ) and the Fourier Stieltjes algebra B ( G ) B\left ( G \right ) always satisfy T ( n ) T\left ( n \right ) for each n = 1 , 2 , … n = 1,2, \ldots .

It is shown that a locally compact group G G is amenable if and only if every derivation of the Fourier algebra A ( G ) A(G) into a Banach A ( G ) A(G) -bimodule is continuous. Also given are necessary and sufficient conditions for A ( G ) A(G) to be weakly amenable.

For G an infinite compact group we show the existence and nonuniqueness of invariant means on the dual of the Fourier algebra. It follows that the space of weakly almost periodic functionals on the Fourier algebra is a proper closed subspace of the dual of the Fourier algebra. We let G denote an infinite compact group and G its dual. We use the notation of [2, Chapters 7 and 8]. Recall A(G) denotes the Fourier algebra of G (an algebra of continuous functions on G) and =Sf °°(G) denotes its dual under the pairing (/, ) (feA(G), $£^{0)). Further note a"°{u) is identified with the C*-algebra of bounded operators on L2(G) commuting with right translation. The module action of A(G) on ??""{o) is defined by the following: for feA(G), e^°°(G), fi 0, (2)p(I)=l (7 is the identity in J^°°(G)), and (3) p(f- )=f(e)p{ || = 1.) If G is also abelian, then G is a discrete group and (3) is equivalent to p( ), by x. Let P={feA(G):f(e)=\ and/is positive definite}. Now £ is a convex spanning subset of A(G) and a commutative semigroup under pointwise multiplication. Further P contains functions with supports in arbitrarily small neighborhoods of e. The collection of invariant means on J^X(G) will be denoted by Jf. Thus for pe^x(G)* (the dual of -2"° ((?)), peJT if and only if (l)p(cp)>0 whenever c^O, (2)p(I)=1, and {3)p(f- ),feP, -/(«), and f-I=f(e)I (feA(G)). In JS?°°(G) let iv* denote the weak-* topology oLx denote the canonical embedding of A(G) intoi?10^)*. Received by the editors October 1, 1970 and, in revised form, October 29, 1970. AMS 1970 subject classifications. Primary 43A75, 43-00.

Let G be a locally compact group with Fourier algebra A ( G ) A(G) . We prove that if G is amenable then every centralizer of A ( G ) A(G) is determined by multiplication with an element of the Fourier-Stieltjes algebra of G. This result is then used to show that isometric centralizers correspond to characters of G.

Let G be a locally compact amenable group. The elements of the Fourier-algebra A (G) are characterized with the aid of the theorem of Schoenberg-Eberlein-Eymard. In this paper an extension of the theorem of Schoenberg-EberleinEymard is shown, which yields a characterization of the Fourieralgebra A(G) of a locally compact amenable group. For abelian groups our result is due to Raouf Doss [2 ]. Let G be a locally compact group with left-invariant Haar measure dx and let L'(G), p _ 1, be the Banach space of p-integrable complexvalued functions on G. By M1(G) we denote the convolution-algebra of the bounded Radon-measures on G. Let p be the left-regular representation and 2 the set of all (classes of) strongly continuous unitary representations of G. It is known that every ireC in the Hilbert space I, can be lifted to a nondegenerate representation of LI(G) by

Let A ( G ) A(G) be the Fourier algebra of a compact group G G . It is shown that a function defined on a closed convex subset of the plane operates in A ( G ) A(G) if and only if it is real analytic. This was shown by Helson, Kahane, Katznelson and Rudin when G G is locally compact and abelian and by Dunkl when G G is compact and contains an infinite abelian subgroup. A direct proof is given of the following lemma which is all that is needed in order to apply the proof of Helson, Kahane, Katznelson and Rudin ( | | | | ||\;|| is the Fourier algebra norm). Lemma. Let r > 0 r > 0 and S r {S_r} be the set of f ∈ A ( G ) f \in A(G) such that f f is real and | | f | | = r ||f|| = r . Then \[ sup f ∈ S r | | e i f | | = e r . \sup \limits _{f \in {S_r}} ||{e^{if}}|| = {e^r}. \]

Let G be a compact nonabelian group and H be a closed subgroup of G. Then H is a set of spectral synthesis for the Fourier algebra A ( G ) A(G) (and indeed for A p ( G ) , 1 ≦ p > ∞ {A^p}(G),1 \leqq p > \infty ). For 1 ≦ p > ∞ 1 \leqq p > \infty , each L p ( G ) {L^p}(G) -multiplier T corresponds to a L p ( H ) {L^p}(H) -multiplier S by the rule ( T f ) | H = S ( f | H ) , f ∈ A ( G ) (Tf)|H = S(f|H),f \in A(G) , if and only if the support of T is contained in H.

Let G G be a compact abelian group, and Γ \Gamma its character group. Given E ⊂ Γ , E a E \subset \Gamma ,{E^a} denotes the set of all accumulation points of E E in Γ ¯ \overline \Gamma , the Bohr compactification of Γ \Gamma . In this paper it is shown that the inclusion ( L 1 ( G ) ) ∧ | E ⊂ ( l 1 ( G ) ) ∧ | E {({L^1}(G))^ \wedge }{|_E} \subset {({l^1}(G))^ \wedge }{|_E} obtains if and only if E ∩ E a = ∅ E \cap {E^a} = \emptyset and there exists a measure μ ϵ M ( G ) \mu \epsilon M(G) such that μ ^ = 1 \widehat \mu = 1 on E E and μ ^ = 0 \widehat \mu = 0 on Γ ∩ E a \Gamma \cap {E^a} .