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Based on Katznelson–Tzafriri Theorem on power bounded operators, we prove in this paper a theorem, which applies to the most of the classical Banach algebras of harmonic analysis associated with locally compact groups, to deal with the problems when a given Banach algebra A is Arens regular and when A is an ideal in its bidual. In the second part of the paper, we study the topological center of the bidual of a class of Banach algebras with a multiplier bounded approximate identity. [ABSTRACT FROM AUTHOR]

In this paper, we shall answer an open question posed in 2015 by X. Chen, A. T.-M. Lau and C.-K. Ng and show that the left amenability of a semitopological semigroup S can be characterized by a fixed point property of affine representations on a dual Banach space. The same machinery can be applied to derive fixed point properties for affine actions of S on dual Banach spaces which characterize the existence of left invariant mean on, respectively, the space of weakly left uniformly continuous functions, the space of almost periodic functions and the space of weakly almost periodic functions on S. [ABSTRACT FROM AUTHOR]

Let M be a semi‐finite von Neumann algebra with normal state space S(M). For any ϕ∈S(M), let Mϕ:={x∈M:xϕ=ϕx} be the centralizer of ϕ with center Z(Mϕ). We show that for ϕ,ψ∈S(M), the following are equivalent. ϕ=ψ.Z(Mψ)⊆Z(Mϕ) and ϕ|Z(Mϕ)=ψ|Z(Mϕ).ϕ,ψ have the same distances to all the closed faces of S(M). As an application, we give an alternative proof of the fact that metric preserving surjections between normal state spaces of semi‐finite von Neumann algebras are induced by Jordan ∗‐isomorphisms between the underlying algebras. We then use it to verify some facts concerning F‐algebras and Fourier algebras of locally compact quantum groups. [ABSTRACT FROM AUTHOR]

Let A be an F -algebra and S (A *) be the metric semi-group of normal states of the W * -algebra A * ⁠. We show that S (A *) determines A , when A * is of type I. In particular, when A is either the Fourier algebra A (G) or the Fourier–Stieltjes algebra B (G) of a type I locally compact group G , the metric semi-group S (A *) determines G up to opposition. A remark for the general case will also be given. [ABSTRACT FROM AUTHOR]

Let G be a locally compact group. Let A be any one of the (complex) Banach algebras: L1(G), M(G), WAP(G) and LUC(G), consisting of integrable functions, regular Borel complex measures, weakly almost periodic functions and bounded left uniformly continuous functions, respectively, on G. We show that the metric semigroup A1+: = {f ∈ A: f ≥ 0 and ∥f∥ = 1} (the convex structure is not considered) is a complete invariant for G. [ABSTRACT FROM AUTHOR]

Abstract: Let G be an admissible compact Hausdorff right topological group, that is, a group with a Hausdorff topology such that for each , the map is continuous, and the set of such that the map is continuous is dense in G. Such groups arise in the study of distal flows. In this paper we study the Fourier–Stieltjes algebra , the linear span of the continuous positive definite functions on G. We show that is isomorphic with the Fourier–Stieltjes algebra of an associated compact topological group. This result is then applied to obtain some geometric properties including the weak and -fixed point properties on . We also study some related properties on the measure algebra . [Copyright &y& Elsevier]

Abstract: This paper is devoted to the study of orthogonality and disjointness preserving linear maps between Fourier and Fourier–Stieltjes algebras of locally compact groups. We show that a linear bijection (resp. ) between two Fourier algebras (resp. Fourier–Stieltjes algebras) of locally compact groups will induce a topological group isomorphism between and , provided that Ψ preserves both disjointness and some kind of orthogonality. This improves earlier results of J.J. Font and M.S. Monfared, where amenability of the groups or continuity of the linear maps are assumed. We also study the structure of bounded and unbounded disjointness preserving linear functionals of Fourier algebras. In the development, general results about disjointness and orthogonality preserving linear maps between -algebras, -algebras and their preduals are obtained. [Copyright &y& Elsevier]

Abstract: In this paper we show that the Fourier–Stieltjes algebra of a non-compact locally compact group G cannot have the fixed point property for nonexpansive mappings. This answers two open problems posed at a conference in Marseille-Luminy in 1989. We also show that a locally compact group is compact exactly if the asymptotic centre of any non-empty closed bounded convex subset C in with respect to a decreasing net of bounded subsets is a non-empty norm compact subset. In particular, when G is compact, has the fixed point property for left reversible semigroups. This generalizes a classical result of T.C. Lim for the circle group. As a consequence of our main results we obtain that a number of properties, some of which were known to hold for compact groups, in fact characterize compact groups. [Copyright &y& Elsevier]

Abstract: In this paper we investigate when various Banach algebras associated to a locally compact group G have the weak or weak∗ fixed point property for left reversible semigroups. We proved, for example, that if G is a separable locally compact group with a compact neighborhood of the identity invariant under inner automorphisms, then the Fourier–Stieltjes algebra of G has the weak∗ fixed point property for left reversible semigroups if and only if G is compact. This generalizes a classical result of T.C. Lim for the case when G is the circle group T. [Copyright &y& Elsevier]