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A Banach algebra \(\mathscr{A}\) is Arens-regular when all its continuous functionals are weakly almost periodic, in symbols when \(\mathscr{A^{∗}} = \mathscr{WAP}(\mathscr{A})\). To identify the opposite behaviour, Granirer called a Banach algebra extremely non-Arens regular (enAr, for short) when the quotient \(\mathscr{A^{∗}} /\mathscr{WAP}(\mathscr{A})\) contains a closed subspace that has \(\mathscr{A^{∗}}\) as a quotient. In this paper we propose a simplification and a quantification of this concept. We say that a Banach algebra \(\mathscr{A}\) is r-enAr, with \(r ≥ 1\), when there is an isomorphism with distortion \(r\) of \(\mathscr{A^{∗}}\) into \(\mathscr{A^{∗}} /\mathscr{WAP}(\mathscr{A})\). When \(r = 1\), we obtain an isometric isomorphism and we say that \(\mathscr{A}\) is isometrically enAr. We then identify sufficient conditions for the predual \(\mathfrak{V_{∗}}\) of a von Neumann algebra \(\mathfrak{V}\) to be r-enAr or isometrically enAr. With the aid of these conditions, the following algebras are shown to be r-enAr: (i) the weighted semigroup algebra of any weakly cancellative discrete semigroup, when the weight is diagonally bounded with diagonal bound \(c ≥ r\). When the weight is multiplicative, i.e., when \(c = 1\), the algebra is isometrically enAr, (ii) the weighted group algebra of any non-discrete locally compact infinite group and for any weight, (iii) the weighted measure algebra of any locally compact infinite group, when the weight is diagonally bounded with diagonal bound \(c ≥ r\). When the weight is multiplicative, i.e., when \(c = 1\), the algebra is isometrically enAr. The Fourier algebra \(A(G)\) of a locally compact infinite group \(G\) is shown to be isometrically enAr provided that (1) the local weight of \(G\) is greater or equal than its compact covering number, or (2) \(G\) is countable and contains an infinite amenable subgroup.

In the mid 1980s it was conjectured that every bispectral meromorphic function ψ ⁢ ( x , y ) {\psi(x,y)} gives rise to an integral operator K ψ ⁢ ( x , y ) {K_{\psi}(x,y)} which possesses a commuting differential operator. This has been verified by a direct computation for several families of functions ψ ⁢ ( x , y ) {\psi(x,y)} where the commuting differential operator is of order ≤ 6 {\leq 6} . We prove a general version of this conjecture for all self-adjoint bispectral functions of rank 1 and all self-adjoint bispectral Darboux transformations of the rank 2 Bessel and Airy functions. The method is based on a theorem giving an exact estimate of the second- and first-order terms of the growth of the Fourier algebra of each such bispectral function. From it we obtain a sharp upper bound on the order of the commuting differential operator for the integral kernel K ψ ⁢ ( x , y ) {K_{\psi}(x,y)} leading to a fast algorithmic procedure for constructing the differential operator; unlike the previous examples its order is arbitrarily high. We prove that the above classes of bispectral functions are parametrized by infinite-dimensional Grassmannians which are the Lagrangian loci of the Wilson adelic Grassmannian and its analogs in rank 2.

We establish some characterizations of the weak fixed point property (weak fpp) for noncommutative (and commutative) L 1 spaces and use this for the Fourier algebra A(G) of a locally compact group G. In particular we show that if G is an IN-group, then A(G) has the weak fpp if and only if G is compact. We also show that if G is any locally compact group, then A(G) has the fixed point property (fpp) if and only if G is finite. Furthermore if a nonzero closed ideal of A(G) has the fpp, then G must be discrete.

The Fourier algebra of the affine group of the real line has a natural identification, as a Banach space, with the space of trace-class operators on $L^2({\mathbb R}^\times, dt/ |t|)$. In this paper we study the "dual convolution product" of trace-class operators that corresponds to pointwise product in the Fourier algebra. Answering a question raised in work of Eymard and Terp, we provide an intrinsic description of this operation which does not rely on the identification with the Fourier algebra, and obtain a similar result for the connected component of this affine group. In both cases we construct explicit derivations on the corresponding Banach algebras, verifying the derivation identity directly without requiring the inverse Fourier transform. We also initiate the study of the analogous Banach algebra structure for trace-class operators on $L^p({\mathbb R}^\times, dt/ |t|)$ for $p\in (1,2)\cup(2,\infty)$., Comment: v2: AMS-LaTeX, 27 pages, 2 figures. One typo fixed following referee's report; hyperref settings tweaked. To appear in Complex Analysis and Operator Theory

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} .

We study locally compact groups for which the Fourier algebra coincides with the Rajchman algebra. In particular, we show that there exist uncountably many non-compact groups with this property. Generalizing a result of Hewitt and Zuckerman, we show that no non-compact nilpotent group has this property, whereas non-compact solvable groups with this property are known to exist. We provide several structural results on groups whose Fourier and Rajchman algebras coincide as well as new criteria for establishing this property. Finally, we study the relation between groups with completely reducible regular representation and groups whose Fourier and Rajchman algebras coincide. For unimodular groups with completely reducible regular representation, we show that the Fourier algebra may in general be strictly smaller than the Rajchman algebra., Comment: 26 pages. Comments welcome

For a Banach function algebra A, we consider the problem of representing a continuous d-homogeneous polynomial P : A → X , where X is an arbitrary Banach space, that satisfies the property P ( f + g ) = P ( f ) + P ( g ) whenever f , g ∈ A are such that supp ( f ) ∩ supp ( g ) = ∅ . We show that such a polynomial can be represented as P ( f ) = T ( f d ) ( f ∈ A ) for some continuous linear map T : A → X for a variety of Banach function algebras such as the algebra of continuous functions C 0 ( Ω ) for any locally compact Hausdorff space Ω, the algebra of Lipschitz functions lip α ( K ) for any compact metric space K and α ∈ ] 0 , 1 [ , the Figa–Talamanca–Herz algebra A p ( G ) for some locally compact groups G and p ∈ ] 1 , + ∞ [ , the algebras A C ( [ a , b ] ) and B V C ( [ a , b ] ) of absolutely continuous functions and of continuous functions of bounded variation on the interval [ a , b ] . In the case where A = C n ( [ a , b ] ) , P can be represented as P ( f ) = ∑ T ( n 1 , … , n d ) ( f ( n 1 ) ⋯ f ( n d ) ) , where the sum is taken over ( n 1 , … , n d ) ∈ Z d with 0 ≤ n 1 ≤ … ≤ n d ≤ n , for appropriate continuous linear maps T ( n 1 , … , n d ) : C n − n d ( [ a , b ] ) → X .

We study the Cauchy problem for Hartree equation with cubic convolution nonlinearity F ( u ) = ( K ∗ | u | 2 ) u under a specified condition on potential K with Cauchy data in modulation spaces M p , q ( R d ) . We establish global well-posedness results in M 1 , 1 ( R d ) when K ( x ) = λ | x | − γ ( λ ∈ R , 0 γ min { 2 , d / 2 } ) ; in M p , q ( R d ) ( 1 ≤ q ≤ min { p , p ′ } where p ′ is the Holder conjugate of p ∈ [ 1 , 2 ] ) when K is in Fourier algebra F L 1 ( R d ) , and local well-posedness result in M p , 1 ( R d ) ( 1 ≤ p ≤ ∞ ) when K ∈ M 1 , ∞ ( R d ) .

We study the Cauchy problem for fractional Schr\"odinger equation with cubic convolution nonlinearity ($i\partial_t u - (-\Delta)^{\frac{\alpha}{2}}u\pm (K\ast |u|^2) u =0$) with Cauchy data in the modulation spaces $M^{p,q}(\mathbb R^{d}).$ For $K(x)= |x|^{-\gamma}$ $ (0< \gamma< \text{min} \{\alpha, d/2\})$, we establish global well-posedness results in $M^{p,q}(\mathbb R^{d}) (1\leq p \leq 2, 1\leq q < 2d/ (d+\gamma))$ when $\alpha =2, d\geq 1$, and with radial Cauchy data when $d\geq 2, \frac{2d}{2d-1}< \alpha < 2. $ Similar results are proven in Fourier algebra $\mathcal{F}L^1(\mathbb R^d) \cap L^2(\mathbb R^d).$, Comment: 15 pages

As is well known, the equivalence between amenability of a locally compact group $G$ and injectivity of its von Neumann algebra $\mathcal{L}(G)$ does not hold in general beyond inner amenable groups. In this paper, we show that the equivalence persists for all locally compact groups if $\mathcal{L}(G)$ is considered as a $\mathcal{T}(L_2(G))$-module with respect to a natural action. In fact, we prove an appropriate version of this result for every locally compact quantum group., 23 pages, updated version

Let G G be a compact connected Lie group. The question of when a weighted Fourier algebra on G G is completely isomorphic to an operator algebra will be investigated in this paper. We will demonstrate that the dimension of the group plays an important role in the question. More precisely, we will get a positive answer to the question when we consider a polynomial type weight coming from a length function on G G with the order of growth strictly bigger than half of the dimension of the group. The case of S U ( n ) SU(n) will be examined, focusing more on the details including negative results. The proof for the positive directions depends on a non-commutative version of the Littlewood multiplier theory, which we will develop in this paper, and the negative directions will be taken care of by restricting to a maximal torus.

We show that an n-homogeneous polynomial P on the Fourier algebra A ( G ) of a locally compact group G can be represented in the form P ( f ) = 〈 T , f n 〉 ( f ∈ A ( G ) ) for some T in the group von Neumann algebra VN ( G ) of G if and only if it is orthogonally additive and completely bounded.

Suppose that f is the characteristic function of a probability measure on the real line \(\mathbb R\). In this paper, we deal with the following problem posed by N.G. Ushakov: Is it true that f is never determined by its imaginary part \(\mathfrak {I}f\)? In other words, is it true that for any characteristic function f there exists a characteristic function g such that \(\mathfrak {I}f\equiv \mathfrak {I}g\) but \( f\not \equiv g\)? We study this question in the more general case of the characteristic function defined on an arbitrary locally compact abelian group. A characterization of what characteristic functions are uniquely determined by their imaginary parts are given. As a consequence of this characterization, we obtain that several frequently used characteristic functions on the classical locally compact abelian groups are uniquely determined by their imaginary parts.

For a locally compact group G and 1 p ∞ let A p ( G ) be the Figa-Talamanca–Herz algebras, which include in particular the Fourier algebra of G, A ( G ) ( p = 2 ). It is shown that for any amenable group H, a proper affine map α : Y ⊂ H → G induces a p-completely contractive algebra homomorphism ϕ α : A p ( G ) → A p ( H ) by setting ϕ α ( u ) = u ∘ α on Y and ϕ α ( u ) = 0 off of Y. Moreover, we show that if both G and H are amenable then any p-completely contractive algebra homomorphism ϕ : A p ( G ) → A p ( H ) is of this form. These results are the analogs in the context of the Figa-Talamanca–Herz algebras of the ones in the Fourier algebra setting ( p = 2 ) initiated by the author and continued with N. Spronk, which in turn generalize results of P.J. Cohen and B. Host from abelian group algebra setting.

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}) .

Let H be an ultraspherical hypergroup associated to a locally compact group G and a spherical projector π (in Muruganandam, Math. Nachr. 281:1590–1603, 2008). We investigate spectral synthesis properties of the Fourier algebra A(H), partly building on results for A(G). Special emphasis is placed on double coset hypergroups. We also present several examples displaying the diverse behavior of hypergroups in contrast to groups.

Let A be a Lau algebra and let S A be the semigroup of all positive functionals in A with norm one. We obtain some equivalent conditions for left amenability of A in terms of a common fixed point property and the Hahn–Banach property. We then apply these results for certain Lau algebras on a locally compact group G to give characterizations for amenability of G .

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

We show that if $G$ is a finite Abelian group and $f$ is an integer-valued map on $G$ with algebra norm at most $M$ then there is some $L < \exp(M^{4+o(1)})$, cosets of (possibly different) subgroups $W_1,...,W_L$, and $s_1,...,s_L \in \{-1,1\}$ such that $f=\sum_i{s_i1_{W_i}}$., 55pp; corrected arXiv title

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 .

We define complete order amenability and first complete order cohomology groups for quantized Banach ordered algebras and show that the vanishing of the latter is equivalent to the operator amenability for the Fourier algebra.

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}.

Let G be a locally compact group and let A(G) and B(G) be the Fourier algebra and the Fourier-Stieltjes algebra of G, respectively. For any unitary representation π of G, let B π (G) denote the w*-closed linear subspace of B(G) generated by all coefficient functions of π, and B 0 π(G) the closure of B π (G) ∩ A c (G), where A c (G) consists of all functions in A(G) with compact support. In this paper we present descriptions of B 0 π(G) and its orthogonal complement B s π(G) in B π (G), generalizing a recent result of T. Miao. We show that for some classes of locally compact groups G, there is a dichotomy in the sense that for arbitrary π, either B 0 π(G) = {0} or B 0 π(G) = A(G). We also characterize functions in B 0 π(G) = A c (G) + B 0 π(G) and study the question of whether B 0 π(G) = A(G) implies that π weakly contains the regular representation.

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 A be a commutative Banach algebra with a BAI (=bounded approximate identity). We equip A** with the (first) Arens multiplication. To each idempotent element u of A** we associate the closed ideal I-u = {a is an element of A : au = 0} in A. In this paper we present a characterization of the closed ideals of A with BAI's in terms of idempotent elements of A**. The main results are: a) A closed ideal I of A has a BAT if there is an idempotent u is an element of A** such that I = I-u and the subalgebra Au is norm closed in A**. b) For any closed ideal I of A with a BAT, the quotient algebra A/I is isomorphic to a subalgebra of A**. We also show that a weak* closed invariant subspace X of the group von Neumann algebra VN(G) of an amenable group G is naturally complemented in VN(G) if the spectrum of X belongs to the closed coset ring R-c(G(d)) of G(d), the discrete version of G. This paper contains several applications of these results., NSERC

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.

Let G be a locally compact group and B(G) the Fourier-Stieltjes algebra of G. We study the problem of how weak8-closedness of some translation invariant subspaces of B(G) is related to the structure of G. Moreover, we prove that for a closed subgroup H of G, the restriction map from B(G) to B(H) is weak8-continuous only when H is open in G. INTRODUCTION Let G be a locally compact group, and let B(G) be the Fourier-Stieltjes algebra of G as defined by Eymard [8]. Recall that B(G) is the linear span of all continuous positive definite functions on G and can be identified with the Banach space dual of C*(G), the group C*-algebra of G. The space B(G), with the xlorm as dual of C*(G), is a commutative Banach *-algebra with pointwise multiplication and complex conjugation. The Fourier algebra A(G) of G is the closed *-subalgebra of B(G) generated by the functions in B(G) with compact support. In particular, A(G) is contained in Co(G), the algebra of complex valued continuous functions on G vanishing at infinity. As is well known A(G) is weak*-dense in B(G) if and only if G is amenable. In [3] translation invariant *-subalgebras A of B(G) were studied, and it was shown that if such A is weak*-closed and point separating, then it must contain A(G). However, apart from this, very little seems to be known about weak*-closed subspaces of B(G). The first purpose of this paper is to investigate the relation between weak*closedness of certain interesting norm-closed translation invariant subspaces of B(G) and the structure of G. Secondly, we solve the problem of whexl, for a closed subgroup H of G, the restriction map from B(G) to B(H) is weak*-continuous. A brief outline of the paper is as follows. In Section 2 we estsablish for almost connected locally compact groups G the relation between weclk*-closed.ness of Bo(G) = B(G) n Co(G) in E3(G) and the structure of G (Theorem 2.10). The key result is that for a connected Lie group G, Bo(G) is weak*-closed in B(G) if and only if G is a reductive Lie group with compact centre and Kazhdan's property (T). Received by the editors December 1S, 1995. 199l Mathematics Subject Classification. Primary 22D10, 43A30. Work supported by NATO collaborative research grant CRG 940184. (g)1997 American Mathematical Society

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.

Necessary and sufficient conditions are given for an ideal I ( H ) I(H) of the Fourier algebra to be complemented when H H is a closed subgroup of G G . Using the Radon Nikodym property, an example of a group G G with a normal abelian subgroup H H for which I ( H ) I(H) is not complemented is presented.

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 G be a locally compact group, A ( G ) A(G) the Fourier algebra of G G , B ( G ) B(G) the Fourier-Stieltjes algebra of G G and VN ( G ) {\text {VN}}(G) the von Neumann algebra generated by the left regular representation λ \lambda of G G . Then A ( G ) A(G) is the predual of VN ( G ) {\text {VN}}(G) ; VN ( G ) {\text {VN}}(G) is a B ( G ) B(G) -module and A ( G ) A(G) is a closed ideal of B ( G ) B(G) . Let AP ( G ^ ) = { T ∈ VN ( G ) : u ↦ u ⋅ T {\text {AP}}(\hat G) = \{ T \in {\text {VN}}(G):u \mapsto u \cdot T is a compact operator from A ( G ) A(G) into VN ( G ) } {\text {VN}}(G)\} , the space of almost periodic operators in VN ( G ) {\text {VN}}(G) . Let C δ ∗ ( G ) C_\delta ^*(G) be the C ∗ {C^*} -algebra generated by { λ ( x ) : x ∈ G } \{ \lambda (x):x \in G\} . Then C δ ∗ ( G ) ⊂ AP ( G ^ ) C_\delta ^*(G) \subset {\text {AP}}(\hat G) . For a compact G G , let E E be the rank one operator on L 2 ( G ) {L^2}(G) that sends h ∈ L 2 ( G ) h \in {L^2}(G) to the constant function ∫ h ( x ) d x \int {h(x)dx} . We have the following results: (1) There exists a compact group G G such that E ∈ AP ( G ^ ) ∖ C δ ∗ ( G ) E \in \text {AP}(\hat G)\backslash C_\delta ^*(G) . (2) For a compact Lie group G G , E ∈ AP( G ^ ) ⇔ E ∈ C δ ∗ ( G ) ⇔ L ∞ ( G ) E \in {\text {AP(}}\hat G{\text {)}} \Leftrightarrow E \in C_\delta ^*(G) \Leftrightarrow {L^\infty }(G) has a unique left invariant mean ⇔ G \Leftrightarrow G is semisimple. (3) If G G is an extension of a locally compact abelian group by an amenable discrete group then AP ( G ^ ) = C δ ∗ ( G ) {\text {AP}}(\hat G) = C_\delta ^*(G) . (4) Let G = F r G = {{\mathbf {F}}_r} , the free group with r r generators, 1 > r > ∞ 1 > r > \infty . If T ∈ VN ( G ) T \in {\text {VN}}(G) and u ↦ u ⋅ T u \mapsto u \cdot T is a compact operator from B ( G ) B(G) into VN ( G ) {\text {VN}}(G) then T ∈ C δ ∗ ( G ) T \in C_\delta ^*(G) .

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.