Representations and positive definite functions on topological semigroups

Abstract

Abstract. A number of theorems are established about positive definite functions and representations of certain topological semigroups. In particular we establish theorems which show that measurable positive definite functions and measurable representations can each be decomposed into the sum of two parts one of which is continuous and the other of which is "small".1. Introduction. In this paper we aim to prove some theorems about positive definite functions, and about the closely related concept of representations, on locally compact topological semigroups; we shall also answer problems raised in [8] and [9] by improving previous theorems on these topics. Theorems about these concepts must, of necessity, apply to the special case of a topological group, and therefore will normally be a "generalization" of a known result for locally compact groups. The theorems we shall establish are true for groups, as a result of the existence of the group algebra, and so our results will often only apply to semigroups in which there is an analogue of the group algebra. Such semigroups are the so-called foundation semigroups; these semigroups form a large family but do not include all commonly met semigroups; the family includes all topological groups and all discrete semigroups, and is closed under products and subsemigroups and (under some restrictions) quotients.Throughout this paper, we shall be concerned with a semigroup S with a weight function w, and with functions and representations that are w-bounded. The original research which led to this paper was an attempt to prove the two main theorems of Section 4, which concern the representations of positive definite functions and representations, respectively, as the sum of two of them, one being continuous and the other being "small", (4.6 and 4.9). In Section 3 we introduce a lemma of Stone-Weierstrass type that is an essential tool for the results of Section 4, and is also used in Section 6. Firstly, Theorem 4.2 provides a "Bochner theorem" for w-bounded, positive definite functions. We also establish Theorems 5.2 and 6.2 which establish necessary and sufficient conditions for the semi-simplicity of the algebra M(S, w) and for the *-semi-simplicity of M(S, w) if S has an involution, respectively.

THE BANACH ALGEBRA ${\cal F}(S,T)$ AND ITS AMENABILITY OF COMMUTATIVE FOUNDATION $*$-SEMIGROUPS $S$ AND $T$

“…Recall that on a Hausdorff locally compact topological semigroup S the space of all measures µ in M (S) (the Banach algebra of all regular complex bounded measures on S with total variation norm) for which the mappings: x → |µ| * δ x and x → δ x * |µ| (δ x denotes by M a (S) (see [1,2,6]). It is well known that M a (S) is a closed two sided L-ideal of M (S).…”