Symmetries of the C*-algebra of a vector bundle

Abstract

We consider $C^*$-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 $C^*$-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 $\mathcal O_E\rtimes G$. If the action is free and rank $E=n$, then we prove that $\mathcal O_E\rtimes G$ 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 $\mathcal O_E\rtimes G$ becomes a continuous field of crossed products $\mathcal O_n\rtimes G$. For transitive actions, we show that $\mathcal O_E\rtimes G$ is Morita-Rieffel equivalent to a graph $C^*$-algebra.