$\mathrm C^*$-algebras of Boolean inverse monoids – traces and invariant means
Documenta mathematica, Tome 21 (2016), pp. 809-840
To a Boolean inverse monoid S we associate a universal C*-algebra CB∗(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 CB∗(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 CB∗(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.
@article{10_4171_dm_546,
author = {Charles Starling},
title = {$\mathrm C^*$-algebras of {Boolean} inverse monoids {\textendash} traces and invariant means},
journal = {Documenta mathematica},
pages = {809--840},
year = {2016},
volume = {21},
doi = {10.4171/dm/546},
url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/546/}
}
Charles Starling. $\mathrm C^*$-algebras of Boolean inverse monoids – traces and invariant means. Documenta mathematica, Tome 21 (2016), pp. 809-840. doi: 10.4171/dm/546
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