research

A Groupoid Approach to Discrete Inverse Semigroup Algebras

Abstract

Let KK be a commutative ring with unit and SS an inverse semigroup. We show that the semigroup algebra KSKS can be described as a convolution algebra of functions on the universal \'etale groupoid associated to SS by Paterson. This result is a simultaneous generalization of the author's earlier work on finite inverse semigroups and Paterson's theorem for the universal CC^*-algebra. It provides a convenient topological framework for understanding the structure of KSKS, including the center and when it has a unit. In this theory, the role of Gelfand duality is replaced by Stone duality. Using this approach we are able to construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups, generalizing the well-studied case of an inverse semigroup with finitely many idempotents. More generally, we describe the irreducible representations of an inverse semigroup SS that can be induced from associated groups as precisely those satisfying a certain "finiteness condition". This "finiteness condition" is satisfied, for instance, by all representations of an inverse semigroup whose image contains a primitive idempotent

    Similar works