Invertibility in groupoid C*-algebras

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 l_2(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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