Symmetries of the C*-algebra of a vector bundle

2019/12/04 by Deaconu, Valentin
#FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.1912.01750

Abstract

We consider C^*-algebras constructed from compact group actions on complex vector bundles E→ X endowed with a Hermitian metric. An action of G by isometries on E→ X induces an action on the C^*-correspondence Γ(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.

Related

• Cuntz--Pimsner algebras of partial automorphisms twisted by vector bundles I: Fixed point algebra, simplicity and the tracial state space
• Groupoid actions on C^*-correspondences
• The C*-algebra of a vector bundle
• A new look at crossed product correspondences and associated C*-algebras
• C^*-algebras associated to homeomorphisms twisted by vector bundles over finite dimensional spaces