Skip to main content
Log in

Inverse semigroups and combinatorial C*-algebras

  • Published:
Bulletin of the Brazilian Mathematical Society, New Series Aims and scope Submit manuscript

Abstract.

We describe a special class of representations of an inverse semigroup S on Hilbert's space which we term tight. These representations are supported on a subset of the spectrum of the idempotent semilattice of S, called the tight spectrum, which is in turn shown to be precisely the closure of the space of ultra-filters, once filters are identified with semicharacters in a natural way. These representations are moreover shown to correspond to representations of the C*-algebra of the groupoid of germs for the action of S on its tight spectrum. We then treat the case of certain inverse semigroups constructed from semigroupoids, generalizing and inspired by inverse semigroups constructed from ordinary and higher rank graphs. The tight representations of this inverse semigroup are in one-to-one correspondence with representations of the semigroupoid, and consequently the semigroupoid algebra is given a groupoid model. The groupoid which arises from this construction is shown to be the same as the boundary path groupoid of Farthing, Muhly and Yeend, at least in the singly aligned, sourceless case.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ruy Exel*.

Additional information

*Partially supported by CNPq.

About this article

Cite this article

Exel*, R. Inverse semigroups and combinatorial C*-algebras. Bull Braz Math Soc, New Series 39, 191–313 (2008). https://doi.org/10.1007/s00574-008-0080-7

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00574-008-0080-7

Keywords:

Mathematical subject classification:

Navigation