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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]

The purpose of this paper is to give an updated survey on various algebraic and analytic properties of semigroups related to fixed point properties of semigroup actions on a non-empty closed convex subset of a Banach space or, more generally, a locally convex topological vector space. [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]

In this paper, we study common fixed point properties for a family of mappings. We present the extensions of Caristi's fixed point theorem and Markov–Kakutani fixed point theorem for a family of set-valued mappings. [ABSTRACT FROM AUTHOR]

Motivated by the beautiful work of M.A. Rieffel (1965) and of M.E. Walter (1974), we obtain characterisations of the Fourier algebra A ( G ) of a locally compact group G in terms of the class of F -algebras (i.e. a Banach algebra A such that its dual A ′ is a W ⁎ -algebra whose identity is multiplicative on A ). For example, we show that the Fourier algebras are precisely those commutative semisimple F -algebras that are Tauberian, contain a nonzero real element, and possess a dual semigroup that acts transitively on their spectrums. Our characterisations fall into three flavours, where the first one will be the basis of the other two. The first flavour also implies a simple characterisation of when the predual of a Hopf–von Neumann algebra is the Fourier algebra of a locally compact group. We also obtain similar characterisations of the Fourier–Stieltjes algebras of G . En route, we prove some new results on the problem of when a subalgebra of A ( G ) is the whole algebra and on representations of discrete groups. [ABSTRACT FROM AUTHOR]

A well-known result of W. Ray asserts that if C is an unbounded convex subset of a Hilbert space, then there is a nonexpansive mapping T : C → C that has no fixed point. In this paper we establish some common fixed point properties for a semitopological semigroup S of nonexpansive mappings acting on a closed convex subset C of a Hilbert space, assuming that there is a point c ∈ C with a bounded orbit and assuming that certain subspace of C b ( S ) has a left invariant mean. Left invariant mean (or amenability) is an important notion in harmonic analysis of semigroups and groups introduced by von Neumann in 1929 [28] and formalized by Day in 1957 [5] . In our investigation we use the notion of common attractive points introduced recently by S. Atsushiba and W. Takahashi. [ABSTRACT FROM AUTHOR]

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 Iu = {a ∊ 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 BAI iff there is an idempotent u ∊ A** such that I = Iu and the subalgebra Au is norm closed in A**. b) For any closed ideal I of A with a BAI, 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) iff the spectrum of X belongs to the closed coset ring Rc(Gd) of Gd, the discrete version of G. This paper contains several applications of these results. [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: This paper is an invitation to the study of the Fourier–Stieltjes algebra , the linear span of the continuous positive definite complex-valued functions on a topological group G. We study , when G has a host algebra or a group -algebra, the analogue of the group -algebra of a locally compact group. Our main challenge is that a topological group G cannot have a positive regular Borel measure which is left translation invariant unless G is locally compact. [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]

Abstract: In this paper, we shall establish a fixed point property on Fréchet spaces for left reversible semitopological semigroups generalizing some classical results. [Copyright &y& Elsevier]

Abstract: In recent years, there have been considerable interests in the study of when a closed convex subset K of a Banach space has the fixed point property, i.e. whenever T is a non-expansive mapping from K into K, then K contains a fixed point for T. In this paper we shall study fixed point properties of semigroups of non-expansive mappings on weakly compact convex subsets of a Banach space (or, more generally, a locally convex space). By considering the classes of bicyclic semigroups we answer two open questions, one posted earlier by the first author in 1976 (Dalhousie) and the other posted by T. Mitchell in 1984 (Virginia). We also provide a characterization for the existence of a left invariant mean on the space of weakly almost periodic functions on separable semitopological semigroups in terms of fixed point property for non-expansive mappings related to another open problem raised by the first author in 1976. [Copyright &y& Elsevier]

Abstract: In this paper, we explore the properties of projections of norm one on general Banach algebras, in particular the relation with conditional expectations for algebras which arise in harmonic analysis. [Copyright &y& Elsevier]

Abstract: The purpose of this paper is to study iterative schemes of Browder and Halpern types for a semigroup of nonexpansive mappings on a compact convex subset of a smooth (and strictly convex) Banach space with respect to a sequence of strongly asymptotic invariant means defined on an appropriate space of bounded real valued functions of the semigroup. Various applications to the additive semigroup of nonnegative real numbers and commuting pairs of nonexpansive mappings are also presented. [Copyright &y& Elsevier]

Abstract: The purpose of this addendum is to clarify two shortcomings in the paper of the title. [Copyright &y& Elsevier]

We introduce the notion of a generalised F-algebra as a Banach algebra A whose Banach space dual is a J B W ⁎ -triple with a certain additional condition on a distinguished character of A. This generalises the well-known notion of F -algebras, which in turn, includes as special cases many important classes of Banach algebras on locally compact groups and semigroups. Our main use of this new notion is to give some axiomatic characterisations of the Fourier algebras on locally compact groups, strengthening our previous work [37]. But we also quickly review some well-known results in F -algebras that can be extended to the context of generalised F -algebras. [ABSTRACT FROM AUTHOR]