Mathematics – Rings and Algebras
Scientific paper
2009-03-20
Mathematics
Rings and Algebras
Scientific paper
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.
Steinberg Benjamin
No associations
LandOfFree
A Groupoid Approach to Discrete Inverse Semigroup Algebras does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with A Groupoid Approach to Discrete Inverse Semigroup Algebras, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and A Groupoid Approach to Discrete Inverse Semigroup Algebras will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-641701