On the Morita Equivalence of Tensor Algebras

Abstract

Our objective is two fold. First, we want to develop a notion of Morita equivalence for C-correspondences that guarantees that if two C-correspondences E and F are Morita equivalent, then the tensor algebras of E and F, TaOEU and TaOFU, are strongly Morita equivalent in the sense of [8], the Toeplitz algebras, TOEU and TOFU, are strongly Morita equivalent in the sense of Rieffel [32] (see [33] also), and the Cuntz‐Pimsner algebras [28], OOEU and OOFU, are strongly Morita equivalent in the same sense. We were inspired by earlier work of Curto, Williams and the first author in [15] and Combes [14]. These papers investigate conditions implying that the crossed product of two C-dynamical systems are strongly Morita equivalent, given that the coefficient algebras are strongly Morita equivalent. Abadie, Eilers, and Exel [1] were similarly inspired and developed a theory of Morita equivalence for special C-correspondences. Our analysis extends theirs but our arguments are different. (See Example 2.5 and Remarks 3.3 and 3.6 for a comparison of their results and ours.) Second, we want to investigate the converse implication. That is, we study the following question: if TaOEU and TaOFU are strongly Morita equivalent in the sense of [8], when are E and F Morita equivalent in the sense we define? For this converse direction, we focus on the tensor algebras rather than the Toeplitz and Cuntz‐Pimsner algebras because very little can be said in the self-adjoint setting. Here, we were inspired by Arveson’s paper [2] in which it is proved that two ergodic measure-preserving transformations are conjugate if and only if certain non-self-adjoint crossed products associated with the transformations are algebraically isomorphic. (See [3] also.) His analysis was further extended by K. Saito and the first author to cover non-self-adjoint crossed products built from automorphisms of general von Neumann algebras. They showed that non-self-adjoint crossed products built from properly outer automorphisms of von Neumann algebras are isomorphic if and only if the automorphisms are conjugate. See [22] and [23]. Our principal result in the converse direction, Theorem 7.2, asserts that if E and F are two aperiodic correspondences, and if TaOEU is strongly Morita equivalent to TaOFU in the sense of [8], then E and F are Morita equivalent. As we shall see in § 5, the notion of aperiodicity is an exact analogue for correspondences of the concept of ‘free action’ in ergodic theory. Free action was a central hypothesis in Arveson’s