Let K be a commutative ring with unit and S an inverse semigroup. We show
that the semigroup algebra KS can be described as a convolution algebra of
functions on the universal \'etale groupoid associated to S 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 C∗-algebra. It
provides a convenient topological framework for understanding the structure of
KS, 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 S 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