2022/01/26 by Neshveyev, Sergey, Schwartz, Gaute · 3 citations
#FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2201.11033
We study the question whether the representations defined by a dense subset of the unit space of a locally compact étale 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))→ 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 étale groupoids, but S is C^*-regular, while T is not.