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.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

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Noncommutative Geometry in NYC

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