Resticted Algebras on Inverse Semigroups Ii, Positive Definite Functions

In [1] we introduced the concept of restricted representations for an inverse semigroup S and studied the restricted forms of some important Banach algebras on S. In this paper, we continue our study by considering the relation between the restricted positive definite functions and retricted representations. In particular, we prove restricted versions of the Godement’s characterization of the positive definite functions of finite support (Theorem 2.1). These results are used in a forthcoming paper to study the restricted forms of the Fourier and Fourier-Stieltjes algebras on an inverse semigroup S [2]. All over this paper, S denotes a unital inverse semigroup with identity 1. Let us remind that an inverse semigroup S is a discrete semigroup such that for each s ∈ S there is a unique element s ∈ S such that sss = s, sss = s. The set E of idempotents of S consists of elements the form ss, s ∈ S. E is a commutative sub semigroup of S. There is a natural order ≤ on E defined by e ≤ f if and only if ef = e. A ∗-representationof S is a pair {π,Hπ} consisting of a (possibly infinite dimensional) Hilbert space Hπ and a map π : S → B(Hπ)