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Studies of various reproducing function systems emphasized the role of translations and the Fourier periodization function. These influenced the development of the concept of dual integrable representations, a large and important class of unitary representations on LCA groups. The key ingredient is the bracket function that enables the explicit description of corresponding cyclic spaces. Since its introduction, the notion was extended to some specific classes of non-abelian groups, and a natural problem emerged, i.e., whether it can be extended to the entire class of (including non-abelian) locally compact groups. In this paper we solve this problem. Interestingly enough, the bracket function (or rather an operator in this case) keeps all of its useful properties, except the Cauchy-Schwarz inequality. Nevertheless, we show that cyclic spaces still can be represented in terms of bracket-weighted spaces. [ABSTRACT FROM AUTHOR]

In this paper, we present a general version of the algebra AM(G) which was introduced by B. Forrest. Indeed, for a faithful commutative Banach algebra A, we embed it in ℳ(A), the multiplier algebra of A, and obtain Banach algebra AM. Then, we study the spaceability of AM− A and AM (G) ℒ ℒA(G). These results give some characterizations of compactness and discreteness of locally compact groups. Also, we show that AM(G) is an ideal in its second dual if and only if G is discrete. Finally, we study the BSE-property of AM(G). [ABSTRACT FROM AUTHOR]

Let A be a Banach algebra and let φ be a non-zero character on A . Suppose that A M is the closure of the faithful Banach algebra A in the multiplier norm. In this paper, topologically left invariant φ -means on A M ∗ are defined and studied. Under some conditions on A , we will show that the set of topologically left invariant φ -means on A ∗ and on A M ∗ have the same cardinality. The main applications are concerned with the quantum group algebra L 1 (G) of a locally compact quantum group G . In particular, we obtain some characterizations of compactness of G in terms of the existence of a non-zero (weakly) compact left or right multiplier on L M 1 (G) or on its bidual in some senses. [ABSTRACT FROM AUTHOR]

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 ( R × , d t / | 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 ( R × , d t / | t |) for p ∈ (1 , 2) ∪ (2 , ∞) . [ABSTRACT FROM AUTHOR]

For an inverse semigroup S with the set of idempotents E, we find necessary and sufficient conditions for the Fourier algebra A(S) to be module amenable, module character amenable, module (operator) biflat and module (operator) biprojective (as l 1 (E) -module). Some examples show that when S is either a bicyclic inverse semigroup or a Brandt inverse semigroup, A(S) is module amenable, module biprojective and module biflat as well. [ABSTRACT FROM AUTHOR]

A locally compact group G is discrete if and only if the Fourier algebra A(G) has a non-zero (weakly) compact multiplier. We partially extend this result to the setting of ultraspherical hypergroups. Let H be an ultraspherical hypergroup and let A(H) denote the corresponding Fourier algebra. We will give several characterizations of discreteness of H in the terms of the algebraic properties of A(H). We also study Arens regularity of closed ideals of A(H). [ABSTRACT FROM AUTHOR]

In this paper we give an example of a Banach algebra A and a closed ideal I of A such that the multiplier algebra of I is equal to A but I does not have any bounded approximate identity. In the case that I has an approximate identity, we give a necessary condition on I for which A = M (I) , where M (I) denotes the multiplier algebra of I. Finally, as a corollary of our results, we show that the Fourier algebra of an amenable group is strictly dense in the Fourier–Stieltjes algebra. [ABSTRACT FROM AUTHOR]

In this paper, we investigate key assumptions on some Banach algebras to have quasi-weak normal structure (resp. quasi-weak ⋆ normal structure) and to prove in particular that fixed point property for Kannan mappings is satisfied on weakly compact convex subsets in this setting. We establish some conditions on a locally compact group G for which the Fourier and Fourier–Stieltjes–Banach algebras A(G) and B(G) have quasi-normal structure. In addition, we give some examples of Banach spaces X such that L (X) (the Banach algebra of bounded linear operators on X) and some of its closed two-sided ideals associated with Fredholm perturbations have fixed point properties. Finally, we give some comments and interesting questions related to this area. [ABSTRACT FROM AUTHOR]

Let A be a complex commutative semisimple Banach algebra. In this paper, we study some ergodic properties of Cesàro bounded multipliers on A. The results are linked to the sets of synthesis and the main applications are concerned with Fourier and Fourier-Stieltjes algebras on locally compact groups. We study also the structure of ideals associated with multipliers of A and A-invariant projections of the dual space of A. Some related problems are also discussed. [ABSTRACT FROM AUTHOR]

For an amenable inverse semigroup S with the set of idempotents E and a minimal idempotent, we explicitly construct a contractive and positive module operator virtual diagonal on the Fourier algebra A(S), as a completely contractive Banach algebra and operator module over ℓ1(E). This generalizes a well known result of Zhong-Jin Ruan on operator amenability of the Fourier algebra of a (discrete) group Ruan (Am J Math 117:1449-1474, 1995). [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We investigate involutions and trivolutions in the second dual of algebras related to a locally compact topological semigroup and the Fourier algebra of a locally compact group. We prove, among the other things, that for a large class of topological semigroups namely, compactly cancellative foundation $$*$$ -semigroup S when it is infinite non-discrete cancellative, $$M_a(S)^{**}$$ does not admit an involution, and $$M_a(S)^{**}$$ has a trivolution with range $$M_a(S)$$ if and only if S is discrete. We also show that when G is an amenable group, the second dual of the Fourier algebra of G admits an involution extending one of the natural involutions of A( G) if and only if G is finite. However, $$A(G)^{**}$$ always admits trivolution. [ABSTRACT FROM AUTHOR]

The notion of reduced synthesis in the context of harmonic analysis on general locally compact groups is introduced; in the classical situation of commutative groups, this notion means that a function f in the Fourier algebra is annihilated by any pseudofunction supported on f (0). A relationship between reduced synthesis and compact synthesis (i.e., the possibility of approximating compact operators by pseudointegral ones without increasing the support) is determined, which makes it possible to obtain new results both in operator theory and in harmonic analysis. Applications to the theory of linear operator equations are also given. [ABSTRACT FROM AUTHOR]

Motivated by and extending a theorem of Reiter on sets of synthesis in $$\mathbb {R}^N$$ , we establish a general result for Fourier algebras of locally compact groups, even in the wider context of regular, semisimple and Tauberian commutative Banach algebras, which contains Reiter's theorem as a special case and explains why it holds. In addition, we give a number of examples and several further results on weak spectral sets. [ABSTRACT FROM AUTHOR]

We introduce a certain property of commutative Banach alge- bras which we call property OB. We prove that every bounded disjointness-preserving linear map from a commutative Banach algebra with the aforesaid property to any semisimple, commutative Banach algebra is a weighted composition map. Further, it is shown that a variety of important Banach algebras in harmonic analysis have the property OB. [ABSTRACT FROM AUTHOR]

We define module operator amenability for completely contractive Banach algebras which are Banach modules over another Banach algebra with compatible actions and study module operator amenability of the Fourier algebra $$A(S)$$ of an inverse semigroup $$S$$ with the set of idempotents $$E$$ , as a module over the semigroup algebra $$\ell ^{1}(E)$$ . We show that when $$E$$ has a minimum element, module operator amenability of $$A(S)$$ is equivalent to amenability of $$S$$ . [ABSTRACT FROM AUTHOR]

Properties of spectral synthesis are exploited to show that, for a large class of commutative hypergroups and for every compact hypergroup, every closed, re exive, left-translation-invariant subspace of L∝(K) is finite- dimensional. Also, we show that, for a class of hypergroups which includes many commutative hypergroups and all Z-hypergroups, every derivation of L1(K) into an arbitrary Banach L1-bimodule is continuous. [ABSTRACT FROM AUTHOR]

We extend the notion of a strong Ditkin set in the dual group for the $${L^1}$$ -algebra of a locally compact abelian group as well as a large number of results for such sets to the setting of a general regular and semisimple commutative Banach algebra and its spectrum. In particular, we study various stability and inheritance properties. Moreover, we present some applications to Fourier algebras of locally compact groups and an example of a compact, infinite double coset hypergroup for which every closed subset is a strong Ditkin set for its Fourier algebra. [ABSTRACT FROM AUTHOR]

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, ). 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. [ABSTRACT FROM AUTHOR]

Let K be a compact subgroup of a locally compact group G. Completely complemented ideals in A( G/ K) are characterised. Biprojectivity and biflatness for the Fourier algebra A( G/ K) are studied. A( G/ K) is operator biprojective precisely when K is open and if this happens, then G does not contain the free group on two generators as a closed subgroup. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We introduce the notion of the Fourier and Fourier-Stieltjes algebras of a topological *-semigroup and show that these are commutative Banach algebras. For a class of foundation semigroups, we show that these are preduals of von Neumann algebras. [ABSTRACT FROM AUTHOR]