Ideal Structure and Simplicity of the C*-Algebras Generated by Hilbert Bimodules

Tsuyoshi Kajiwara, Claudia Pinzari, Yasuo Watatani

Research output: Contribution to journalArticle

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Abstract

Pimsner introduced the C*-algebra OXgenerated by a Hilbert bimoduleXover a C*-algebra A. We look for additional conditions thatXshould satisfy in order to study the simplicity and, more generally, the ideal structure of OXwhenXis finite projective. We introduce two conditions, "(I)-freeness" and "(II)-freeness," stronger than the former, in analogy with J. Cuntz and W. Krieger (Invent. Math.56, 1980, 251-268) and J. Cuntz (Invent. Math.63, 1981, 25-40), respectively. (I)-freeness comprehends the case of the bimodules associated with an inclusion of simple C*-algebras with finite index, real or pseudoreal bimodules with finite intrinsic dimension, and the case of "Cuntz-Krieger bimodules." IfXsatisfies this condition the C*-algebra OXdoes not depend on the choice of the generators when A is faithfully represented. As a consequence, ifXis (I)-free and A isX-simple, then OXis simple. In the case of Cuntz-Krieger algebras OA,X-simplicity corresponds to the irreducibility of the matrix A. If A is simple and p.i. then OXis p.i.; if A is nonnuclear then OXis nonnuclear. Thus we provide many examples of (purely) infinite nonnuclear simple C*-algebras. Furthermore ifXis (II)-free, we determine the ideal structure of OX.

Original languageEnglish
Pages (from-to)295-322
Number of pages28
JournalJournal of Functional Analysis
Volume159
Issue number2
DOIs
Publication statusPublished - Nov 10 1998

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Bimodule
C*-algebra
Hilbert
Simple C*-algebras
Simplicity
Cuntz-Krieger Algebra
Irreducibility
Analogy
Inclusion
Generator

All Science Journal Classification (ASJC) codes

  • Analysis

Cite this

Ideal Structure and Simplicity of the C*-Algebras Generated by Hilbert Bimodules. / Kajiwara, Tsuyoshi; Pinzari, Claudia; Watatani, Yasuo.

In: Journal of Functional Analysis, Vol. 159, No. 2, 10.11.1998, p. 295-322.

Research output: Contribution to journalArticle

Kajiwara, Tsuyoshi ; Pinzari, Claudia ; Watatani, Yasuo. / Ideal Structure and Simplicity of the C*-Algebras Generated by Hilbert Bimodules. In: Journal of Functional Analysis. 1998 ; Vol. 159, No. 2. pp. 295-322.
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