On the Morita equivalence of tensor algebras. (Q2766361)

The first objective of the paper is to develop a notion of Morita equivalence for \(C^*\)-corresponden\-ces that guarantees that if two \(C^*\)-correspondences \(E\) and \(F\) are Morita equivalent, then the tensor algebras of \(E\) and \(F\), \({\mathcal T}_+(E)\) and \({\mathcal T}_+(F)\), are strongly Morita equivalent in the sense of \textit{D. Blecher}, \textit{P. Muhly} and \textit{V. Paulsen} [Mem. Am. Math. Soc. 681 (2000; Zbl 0966.46033)], the Toeplitz algebras, \({\mathcal T}(E)\) and \({\mathcal T}(F)\), are strongly Morita equivalent in the sense of \textit{M. A. Rieffel} [J. Pure Appl. Algebra 5, 51--96 (1974; Zbl 0295.46099)], and the Cuntz-Pimsner algebras [\textit{M. Pimsner}, Free Probability theory D. Voiculescu (ed.), Fields Inst. Commun. 12, 189--212 (1997; Zbl 0871.46028)], \({\mathcal O}(E)\) and \({\mathcal O}(F)\), are strongly Morita equivalent in the same sense. The second aim is to investigate the converse implication, i.e., if \({\mathcal T}_+(E)\) and \({\mathcal T}_+(F)\) are strongly Morita equivalent, when are \(E\) and \(F\) Morita equivalent in the sense defined by the authors?