Journal of Operator Theory

Volume 64, Issue 2, Fall 2010 pp. 417-434.

Extension of automorphisms to $C^*$ crossed products with non-trivial centre

Authors: Roberto Conti (1) and Ezio Vasselli (2)
Author institution: (1) School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan, NSW 2308, Australia
(2) Dipartimento di Matematica, Universita di Roma ''La Sapienza'', Piazzale Aldo Moro 5, I-00185 Roma, Italy

Summary: Given a quasi-special endomorphism $\rho$ of a $C^*$-algebra $\mathcal{A}$ with nontrivial center, we study an extension problem for automorphisms of $\mathcal{A}$ to a minimal cross-product $\mathcal{B}$ of $\mathcal{A}$ by $\rho$. Exploiting some aspects of the underlying generalized Doplicher-Roberts duality theory based on Pimsner algebras, an obstruction to the existence of such extensions is found and described in terms of sections of a suitable group bundle.

Keywords: $C^*$-algebra, cross-product, automorphism, topological group, duality, tensor $C^*$-category, bimodule, Pimsner algebra, group bundle

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