To a Boolean inverse monoid we associate a universal C*-algebra and show that it is equal to Exel's tight C*-algebra of . We then show that any invariant mean on (in the sense of Kudryavtseva, Lawson, Lenz and Resende) gives rise to a trace on , and vice-versa, under a condition on equivalent to the underlying groupoid being Hausdorff. Under certain mild conditions, the space of traces of is shown to be isomorphic to the space of invariant means of . 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 .
Cite this article
Charles Starling, -algebras of Boolean inverse monoids -- traces and invariant means. Doc. Math. 21 (2016), pp. 809–840DOI 10.4171/DM/546