TY - JOUR

T1 - Ideal Structure and Simplicity of the C*-Algebras Generated by Hilbert Bimodules

AU - Kajiwara, Tsuyoshi

AU - Pinzari, Claudia

AU - Watatani, Yasuo

N1 - Copyright:
Copyright 2004 Elsevier Science B.V., Amsterdam. All rights reserved.

PY - 1998/11/10

Y1 - 1998/11/10

N2 - 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.

AB - 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.

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U2 - 10.1006/jfan.1998.3306

DO - 10.1006/jfan.1998.3306

M3 - Article

AN - SCOPUS:0002283588

VL - 159

SP - 295

EP - 322

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 2

ER -