C\mathrm 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
$\mathrm C^*$-algebras of Boolean inverse monoids -- traces and invariant means cover
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Abstract

To a Boolean inverse monoid SS we associate a universal C*-algebra CB(S)C^*_{B}(S) and show that it is equal to Exel's tight C*-algebra of SS. We then show that any invariant mean on SS (in the sense of Kudryavtseva, Lawson, Lenz and Resende) gives rise to a trace on CB(S)C^*_{B}(S), and vice-versa, under a condition on SS equivalent to the underlying groupoid being Hausdorff. Under certain mild conditions, the space of traces of CB(S)C^*_{B}(S) is shown to be isomorphic to the space of invariant means of SS. 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 SS.

Cite this article

Charles Starling, C\mathrm C^*-algebras of Boolean inverse monoids -- traces and invariant means. Doc. Math. 21 (2016), pp. 809–840

DOI 10.4171/DM/546