# A curious partition of unity argument related to actions
of groupoids and inverse semigroups on C*-algebras

### John Quigg

### Mathematics Department

Arizona State University

### Thursday, January 29, 1998

4:00 PM

### Room 102, Bradley Hall

The study of group actions on C*-algebras ("noncommutative dynamics")
has been a central theme of C*-theory for many years now. But some C*-algebras
(Cuntz-Krieger algebras, for example) don't quite fit into the framework
of group actions---it's necessary to allow actions by "group-like"
objects, including groupoids and inverse semigroups. In joint work with
Nandor Sieben, we've recently been able to show a strong correspondence
between actions of r-discrete groupoids and of inverse semigroups. I'll
focus upon a "curious" (to me, anyway) partition of unity argument
we had to scrape together to show an isomorphism of the associated crossed
product C*-algebras. This argument has nothing to do with C*-algebras per
se, and can be regarded as an answer to the question, "How much collapsing
occurs when we identify functions which agree on overlaps?" Time permitting,
I'll indicate how this leads to the crossed product isomorphism.