Ideal Structure and Simplicity of the C*-Algebras Generated by Hilbert Bimodules
Ideal Structure and Simplicity of the C*-Algebras Generated by Hilbert Bimodules
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DOI：
10.1006/jfan.1998.3306
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发表时间：
1998-02
影响因子：
1.7
通讯作者：
Tsuyoshi Kajiwara;C. Pinzari;Yasuo Watatani Department of Environmental;M. Sciences;Okayama University Dipartimento di Matematica;Universita' di RomaTor Vergata;Graduate Scool of Mathematics;Kyushu University
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作者：
Tsuyoshi Kajiwara;C. Pinzari;Yasuo Watatani Department of Environmental;M. Sciences;Okayama University Dipartimento di Matematica;Universita' di RomaTor Vergata;Graduate Scool of Mathematics;Kyushu University
Pimsner 在 C* 代数 A 上引入了由希尔伯特双模 X 生成的 C* 代数 OX。我们寻找 X 应该满足的附加条件，以便研究当 X 是有限射影时 OX 的简单性和更一般的理想结构。我们引入两个条件，“(I)-自由度”和“(II)-自由度”，比前者更强，分别与 J. Cuntz 和 W. Krieger (Invent. Math.56, 1980, 251–268) 和 J. Cuntz (Invent. Math.63, 1981, 25–40) 类比。 (I)-freeness 理解与包含具有有限索引的简单 C*-代数、具有有限内在维数的实数或伪实数双模以及“Cuntz-Krieger 双模”的情况相关的双模情况。如果X满足这个条件，则当A被忠实地表示时，C*代数OX不依赖于生成器的选择。因此，如果 X 是 (I) 自由的且 A 是 X 简单的，则 OX 是简单的。在 Cuntz-Krieger 代数 OA 的情况下，X-简单性对应于矩阵 A 的不可约性。如果 A 是简单的并且 p.i.然后 OX 为 p.i.；如果 A 是非核的，则 OX 是非核的。因此，我们提供了许多（纯）无限非核简单 C* 代数的例子。此外，如果 X 不含 (II)，我们就确定了 OX 的理想结构。
Pimsner introduced theC*-algebra OXgenerated by a Hilbert bimoduleXover aC*-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 simpleC*-algebras with finite index, real or pseudoreal bimodules with finite intrinsic dimension, and the case of “Cuntz–Krieger bimodules.” IfXsatisfies this condition theC*-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 matrixA. 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 simpleC*-algebras. Furthermore ifXis (II)-free, we determine the ideal structure of OX.