On the Morita Equivalence of Tensor Algebras
by
Baruch Solel
Technion - Israel Institute of Technology
Coauthors: P. S. Muhly

We recall the construction and basic properties of Tensor algebras and C^*-algebras associated with C^*-correspondences (namely, the Cuntz-Pimsner algebra and the associated Toeplitz algebra).

We note that, if the C^*-correspondence over the C^*-algebra A is A itself with a left action defined by an automorphism of A then the Cuntz-Pimsner algebra is the crossed product of A by this automorphism and the Tensor algebra is the analytic (or semi-) crossed product.

We develop a notion of Morita equivalence for general C^*-correspondences and show that if two correspondences are Morita equivalent, then the tensor algebras built from them are strongly Morita equivalent (in the sense of Blecher-Muhly-Paulsen). Also, the Toeplitz algebras are strongly Morita equivalent in the sense of Rieffel, as are the Cuntz -Pimsner algebras. These results extend results of Combes and Curto-Muhly-Williams (for the C^*-crossed product case).

Conversely, if the tensor algebras are strongly Morita equivalent, and if the correspondences are aperiodic in a certain sense, then the correspondences are Morita equivalent.

This generalizes a theorem of Arveson. The notion of aperiodicity, which generalizes the concept of full Connes spectrum for automorphisms, is explored; its role in the ideal theory of tensor algebras and in the theory of their automorphisms is investigated.

Date received: April 11, 1999