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.
“…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].…”
Let S be a foundation locally compact topological semigroup, and let Ma(S) be the space of all measures μ ∈ M (S) for which the maps x → |μ| * δx and x → |μ| * δx from S into M (S) are weakly continuous. The purpose of this article is to develop a notion of character amenability for semigroup algebras. The main results concern the χ-amenability of Ma(S). We give necessary and sufficient conditions for the existence of a left χ-mean on Ma(S) * .
“…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].…”
Let S be a foundation locally compact topological semigroup, and let Ma(S) be the space of all measures μ ∈ M (S) for which the maps x → |μ| * δx and x → |μ| * δx from S into M (S) are weakly continuous. The purpose of this article is to develop a notion of character amenability for semigroup algebras. The main results concern the χ-amenability of Ma(S). We give necessary and sufficient conditions for the existence of a left χ-mean on Ma(S) * .
“…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).…”
The relation between representations and positive definite functions is a key concept in harmonic analysis on topological groups. Recently this relation has been studied on topological groupoids. In this paper, we investigate the concept of restricted positive definite functions and their relation with restricted representations of an inverse semigroup. We also introduce the restricted Fourier and Fourier-Stieltjes algebras of an inverse semigroup and study their relation with the corresponding algebras on the associated groupoid.
“…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].…”
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