Fourier Algebras on Topological Foundation *-Semigroups.

Abstract

We introduce the notion of the Fourier and Fouier-Stieltjes algebra 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.

On character amenability of semigroup algebras

“…Let M (S) be the Banach algebra of all bounded regular Borel measures on S with norm and convolution product [5]. Recall that M a (S) denotes the space of all measures μ ∈ M (S) on S for which the mappings x → δ x * |μ| and x → |μ| * δ x (where δ x denotes the point mass at x for x ∈ S) from S into M (S) are weakly continuous (see [1], [11] and [19]). This algebra was first studied in the sequence of papers by A. C. and J. W. Baker, [3] and [4].…”

Restricted Algebras on Inverse Semigroups—Part II: Positive Definite Functions

“…Then S is a zero inverse semigroup with identity. Here S r = S, as sets, P (S) = {u : u ≥ 0, u is decreasing } [3], but the constant function u = 1 is in P r (S). This in particular shows that the map τ is not necessarily surjective, if S happens to have a zero element.…”

“…In [3] the authors developed harmonic analysis on topological foundation *semigroups (which include all inverse semigroups) and in particular studied positive definite functions on them. Our aim in this section is to develop a parallel theory for the restricted case, and among other results prove the generalization of the Godement's characterization of positive definite functions on groups [4] in our restricted context(Theorem 2.1).…”

Ergodic theory of amenable semigroup actions

“…We recall that, for a semigroup S, M a (S) is the set of all measures µ ∈ M(S) such that both mappings s → |µ| * δ s and s → δ s * |µ| from S into M(S) are weakly continuous (see [1] and [5] for definition). A semigroup S is called a foundation semigroup if {supp µ; µ ∈ M a (S)} is dense in S. It is known that M a (S) admits a positive approximate identity with norm 1 [5].…”