Cuntz-Krieger uniqueness theorem for crossed products by Hilbert bimodules

Abstract

It is shown that a C*-algebra generated by any faithful covariant representation of a Hilbert bimodule X is canonically isomorphic to the crossed product associated to X provided that Rieffel's induced representation functor X-ind is topologically free. It is discussed how this result could be applied to universal C*-algebras generated by relations with a circle gauge action. In particular, it leads to generalizations of isomorphism theorems for various crossed products, and is shown to be equivalent to Cuntz-Krieger uniqueness theorem for finite graph C*-algebras (on that occasion an intriguing realization of Cuntz-Krieger algebras as crossed products by Exel's interactions is discovered).