Opposite algebras of groupoid C*-algebras

TL;DR

This paper proves that all groupoid C*-algebras are isomorphic to their opposites and explores the implications for the classification of C*-algebras, including the existence of non-groupoid C*-algebras.

Contribution

It establishes the isomorphism between groupoid C*-algebras and their opposites and analyzes the structure of section algebras of Fell-bundles, revealing new relationships.

Findings

Every groupoid C*-algebra is isomorphic to its opposite.

Existence of C*-algebras not stably isomorphic to any groupoid C*-algebra.

Opposite algebra of a section algebra of a Fell-bundle is isomorphic to the section algebra of an opposite bundle.

Abstract

We show that every groupoid C*-algebra is isomorphic to its opposite, and deduce that there exist C*-algebras that are not stably isomorphic to groupoid C*-algebras, though many of them are stably isomorphic to twisted groupoid C*-algebras. We also prove that the opposite algebra of a section algebra of a Fell-bundle over a groupoid is isomorphic to the section algebra of a natural opposite bundle.