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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]

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: 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]

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]