Invertibility in groupoid C*-algebras

TL;DR

This paper characterizes invertibility in groupoid C*-algebras by linking it to regular representations at units, and establishes the existence of a unit where the norm of an element matches its regular representation norm.

Contribution

It provides a necessary and sufficient condition for invertibility in groupoid C*-algebras and proves the existence of a unit achieving the norm of elements.

Findings

Invertibility of a in C*(G) iff all regular representations are invertible.

Existence of a unit x with ||a|| = ||λ_x(a)|| for any a in C*(G).

Characterization of invertibility in terms of regular representations.

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 \lambda_x(a) is invertible for every x in the unit space of G, where \lambda_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|| = ||\lambda_x(a)||.