Premium
On the Morita Equivalence of Tensor Algebras
Author(s) -
Muhly Paul S.,
Solel Baruch
Publication year - 2000
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611500012405
Subject(s) - morita equivalence , mathematics , morita therapy , automorphism , pure mathematics , totally disconnected space , tensor (intrinsic definition) , equivalence (formal languages) , algebra over a field , locally compact space
We develop a notion of Morita equivalence for general C * ‐correspondences over C * ‐algebras. We show that if two correspondences are Morita equivalent, then the tensor algebras built from them are strongly Morita equivalent in the sense developed by Blecher, Muhly and Paulsen. Also, the Toeplitz algebras are strongly Morita equivalent in the sense of Rieffel, as are the Cuntz–Pimsner algebras. Conversely, if the tensor algebras are strongly Morita equivalent, and if the correspondences are aperiodic in a fashion that generalizes the notion of aperiodicity for automorphisms of C * ‐algebras, then the correspondences are Morita equivalent. This generalizes a venerated theorem of Arveson on algebraic conjugacy invariants for ergodic, measure‐preserving transformations. The notion of aperiodicity, which also 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. 1991 Mathematics Subject Classification : 46H10, 46H20, 46H99, 46M99, 47D15, 47D25.