C*-algebras of Boolean inverse monoids - traces and invariant means

Abstract

To a Boolean inverse monoid $S$ we associate a universal C*-algebra $C_B^{\ast }(S)$ 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 $C_B^{\ast }(S)$, and vice-versa, under a condition on $S$ equivalent to the underlying groupoid being Hausdorff. Under certain mild conditions, the space of traces of $C_B^{\ast }(S)$ 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 $S$.

29 pages, 1 figure. v3 fixes typos and adds references. This version to appear in Documenta Math

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• A non-amenable groupoid whose maximal and reduced $C^{\ast }$-algebras are the same
• $C^{\ast }$-algebras associated to Boolean dynamical systems
• Some remarks about the weak containment property for groupoids and semigroups
• Partial transformation groupoids attached to graphs and semigroups
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Cited by in corpus (2)

• The dynamics of partial inverse semigroup actions
• Amenability and paradoxicality in semigroups and C*-algebras