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 - Funding Information:
Pimsner introduced the C*-algebra OX generated by a Hilbert bimodule X over a C*-algebra A. We look for additional conditions that X should satisfy in order to study the simplicity and, more generally, the ideal structure of OX when X is 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.’’ If X satisfies this condition the C*-algebra OX does not depend on the choice of the generators when A is faithfully represented. As a consequence, if X is (I)-free and A is X-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 OX is p.i.; if A is nonnuclear then OX is nonnuclear. Thus we provide many examples of (purely) infinite nonnuclear simple C*-algebras. Furthermore if X is (II)-free, we determine the ideal structure of OX. 1998 Academic Press * Supported by the Grants-in-aid for Scientific Research, The Ministry of Science, and Culture, Japan. -Supported by MURST, CNR-GNAFA, and the European Community.

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

SN - 0022-1236

VL - 159

SP - 295

EP - 322

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 2

ER -