Symmetries of the C*-algebra of a vector bundle

Abstract

We consider $C^{\ast }$-algebras constructed from compact group actions on complex vector bundles $E\to X$ endowed with a Hermitian metric. An action of $G$ by isometries on $E\to X$ induces an action on the $C^{\ast }$-correspondence $\Gamma (E)$ over $C(X)$ consisting of continuous sections, and on the associated Cuntz-Pimsner algebra ${\mathcal{O}}_E$, so we can study the crossed product ${\mathcal{O}}_E⋊G$. If the action is free and rank $E=n$, then we prove that ${\mathcal{O}}_E⋊G$ is Morita-Rieffel equivalent to a field of Cuntz algebras ${\mathcal{O}}_n$ over the orbit space $X/G$. If the action is fiberwise, then ${\mathcal{O}}_E⋊G$ becomes a continuous field of crossed products ${\mathcal{O}}_n⋊G$. For transitive actions, we show that ${\mathcal{O}}_E⋊G$ is Morita-Rieffel equivalent to a graph $C^{\ast }$-algebra.