C*-Algebras of Boolean Inverse Monoids -- Traces and Invariant Means

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 $S$.

2010 Mathematics Subject Classification: 20M18, 46L55, 46L05

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