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.
In this paper, we characterize pseudo-contractibility of 1 (S), where S is a uniformly locally finite inverse semigroup. As a consequence, we show that for a Brandt semigroup S = M 0 (G, I), the semigroup algebra 1 (S) is pseudo-contractible if and only if G and I are finite. Moreover, we study the notions of pseudo-amenability and pseudo-contractibility of a semigroup algebra 1 (S) in terms of the amenability of S.
Mathematics Subject Classification (2000). Primary 43A20, 20M18; Secondary 16E40.
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