# $C_{∗}$-algebras of Boolean inverse monoids -- traces and invariant means

### Charles Starling

University of Ottawa Department of Mathematics and Statistics 585 King Edward Ottawa, ON, Canada K1N 6N5

## Abstract

To a Boolean inverse monoid $S$ we associate a universal C*-algebra $C_{B}(S)$ and show that it is equal to Exel's tight C*-algebra of $S$. We then show that any invariant mean on $S$ (in the sense of Kudryavtseva, Lawson, Lenz and Resende) gives rise to a trace on $C_{B}(S)$, and vice-versa, under a condition on $S$ equivalent to the underlying groupoid being Hausdorff. Under certain mild conditions, the space of traces of $C_{B}(S)$ is shown to be isomorphic to the space of invariant means of $S$. We then use many known results about traces of C*-algebras to draw conclusions about invariant means on Boolean inverse monoids; in particular we quote a result of Blackadar to show that any metrizable Choquet simplex arises as the space of invariant means for some AF inverse monoid $S$.

## Cite this article

Charles Starling, $C_{∗}$-algebras of Boolean inverse monoids -- traces and invariant means. Doc. Math. 21 (2016), pp. 809–840

DOI 10.4171/DM/546