Restricted algebras on inverse semigroups I, representation theory

Abstract

Abstract. 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. This is the first in a series of papers in which we have investigated a similar relation on inverse semigroups. We use a new concept of "restricted" representations and study the restricted semigroup algebras and corresponding C * -algebras.

Figa-Talamanca–Herz algebras for restricted inverse semigroups and Clifford semigroups

“…π p (st) does not necessarily coincide with π p (s)π p (t) for all s, t ∈ S. Remark 2.1. As in [1] if we adjoin a zero 0 to S and put 0 * = 0, then we get an inverse semigroup S r with the product •:…”

“…Our aim in this section is to define a new representation of S not only on Hilbert spaces but also on reflexive Banach spaces (cf. [6] for the group case) and then extend it to a representation of l 1 r (S), the restricted semigroup algebra of S. This representation generalizes the * -representation of S defined on l 2 (S) in [1]. Definition 2.1.…”

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

“…Of course, one can always adjoin a unit 1 to T with 1 * 1 to get a unital T . However, positive definite functions on T do not necessarily extend to positive definite functions on T 1 . Following 3 , we consider the subset P e T of extendible positive definite functions on T which are those u ∈ P T such that u u, and there exists a constant c > 0 such that for all n ≥ 1, x 1 , .…”

“…Also the linear span B e T of P e T is an algebra 3, 3.4 which coincides with the set of coefficient functions of * -representations of T 3, 3.2 . If T has a zero element, then so is T 1 . In this case, we put P 0 T {u ∈ P T : u 0 0} and P 0,e T P 0 T ∩ P e T .…”

“…However this case never happens for the directed graph E constructed above, as it has no loops when xx * e or x * x e implies that x e, and has no sink when xx * e implies that x e, for each x ∈ S and e ∈ E, but these two conditions are International Journal of Mathematics and Mathematical Sciences 21 clearly equivalent, and both are equivalent to S E. A concrete example is S N, max with n n * . Also a sufficient condition for E to be cofinal is that S is finitely transitive; namely, for each e, f ∈ E there are finitely many x i ∈ S, 1 ≤ i ≤ n with s x 1 e, r x n f and r x i s x i 1 , for 1 ≤ i ≤ n − 1. Let us say that S is N-transitive if we could always find such a finite path with n ≤ N. A concrete example of a 1-transitive semigroup is the Brandt semigroup B 2 consisting of all pairs i, j , i, j ∈ {0, 1}, plus zero element, with i, j k, l i, l if j k, and zero otherwise.…”