Geodesic
$\mathrm C^*$-algebras of Boolean inverse monoids -- traces and invariant means
Documenta mathematica, Tome 21 (2016), pp. 809-840.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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 $S$.
Classification : 20M18, 46L55, 46L05 
@article{DOCMA_2016__21__a22,
     author = {Starling, Charles},
     title = {$\mathrm C^*$-algebras of {Boolean} inverse monoids -- traces and invariant means},
     journal = {Documenta mathematica},
     pages = {809--840},
     publisher = {mathdoc},
     volume = {21},
     year = {2016},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DOCMA_2016__21__a22/}
}
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Starling, Charles. $\mathrm C^*$-algebras of Boolean inverse monoids -- traces and invariant means. Documenta mathematica, Tome 21 (2016), pp. 809-840. http://geodesic.mathdoc.fr/item/DOCMA_2016__21__a22/