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Morita Equivalence of Operator Algebras and Their C *‐Envelopes
Author(s) -
Blecher David P.,
Muhly Paul S.,
Na Qiyuan
Publication year - 1999
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609399005962
Subject(s) - morita equivalence , mathematics , bimodule , bijection , pure mathematics , operator algebra , noncommutative geometry , operator (biology) , algebra over a field , discrete mathematics , biochemistry , chemistry , repressor , transcription factor , gene
If two operator algebras A and B are strongly Morita equivalent (in the sense of [ 5 ]), then their C *‐envelopes C *( A ) and C *( B ) are strongly Morita equivalent (in the usual C *‐algebraic sense due to Rieffel). Moreover, if Y is an equivalence bimodule for a (strong) Morita equivalence of A and B , then the operation, Y ⊗ hA −, of tensoring with Y , gives a bijection between the boundary representations of C *( A ) for A and the boundary representations of C *( B ) for B . Thus the ‘noncommutative Choquet boundaries’ of Morita equivalent A and B are the same. Other important objects associated with an operator algebra are also shown to be preserved by Morita equivalence, such as boundary ideals, the Shilov boundary ideal, Arveson's property of admissability, and the lattice of C *‐algebras generated by an operator algebra. 1991 Mathematics Subject Classification 47D25, 46L05, 46M99, 16D90.