Invertibility in groupoid C*-algebras

2013/02/07 by Ruy Exel, Exel, Ruy
Mathematics · #22A22 #46L05 #46L55 #FOS: Mathematics #Operator Algebras (math.OA) #math.OA #msc:22A22 #msc:46L05 #msc:46L55

paper · pdf · doi:10.48550/arxiv.1302.1728

8 pages

arxiv created 2013/02/07 · arxiv updated 2013/02/08

Abstract

Given a second-countable, Hausdorff, étale, amenable groupoid G with compact unit space, we show that an element a in C*(G) is invertible if and only if λ_x(a) is invertible for every x in the unit space of G, where λ_x refers to the "regular representation" of C*(G) on l₂(G_x). We also prove that, for every a in C*(G), there exists some x in G⁽⁰⁾ such that ||a|| = ||λ_x(a)||.

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• A non-amenable groupoid whose maximal and reduced C^*-algebras are the same