Symmetries of the C⁎-algebra of a vector bundle

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 OE, so we can study the crossed product OE⋊G. If the action is free and rank E=n, then we prove that OE⋊G is Morita–Rieffel equivalent to a field of Cuntz algebras On over the orbit space X/G. If the action is fiberwise, then OE⋊G becomes a continuous field of crossed products On⋊G. For transitive actions, we show that OE⋊G is Morita–Rieffel equivalent to a graph C⁎-algebra.