Non-Hausdorff etale groupoids and C*-algebras of left cancellative monoids
| Autor: | Neshveyev, Sergey, Schwartz, Gaute |
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| Rok vydání: | 2022 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | We study the question whether the representations defined by a dense subset of the unit space of a locally compact \'etale groupoid are enough to determine the reduced norm on the groupoid C$^*$-algebra. We present sufficient conditions for either conclusion, giving a complete answer when the isotropy groups are torsion-free. As an application we consider the groupoid $G(S)$ associated to a left cancellative monoid $S$ by Spielberg and formulate a sufficient condition, which we call C$^*$-regularity, for the canonical map $C^*_r(G(S))\to C^*_r(S)$ to be an isomorphism, in which case $S$ has a well-defined full semigroup C$^*$-algebra $C^*(S)=C^*(G(S))$. We give two related examples of left cancellative monoids $S$ and $T$ such that both are not finitely aligned and have non-Hausdorff associated \'etale groupoids, but $S$ is C$^*$-regular, while $T$ is not. Comment: 23 pages; v3: minor changes, more references, last section split in two v2: typos corrected, minor changes |
| Databáze: | arXiv |
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