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