Jones index theory for Hilbert C*-bimodules and its equivalence with conjugation theory
Jones index theory for Hilbert C*-bimodules and its equivalence with conjugation theory
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DOI：
10.1016/j.jfa.2003.09.008
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发表时间：
2003-01
影响因子：
1.7
通讯作者：
Tsuyoshi Kajiwara;C. Pinzari;Y. Watatani
中科院分区：
文献类型：
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作者：
Tsuyoshi Kajiwara;C. Pinzari;Y. Watatani
基于一个Pimsner-Popa型不等式，我们在具有双∗结构的C-Hilbertian双模AXB上引入了有限右(左)数值指标的概念。X的右指标可以在A的包络von Neumann代数的中心构造。如果右指数位于A的乘子代数中，则称双模X为有限右指数。在这种情况下，Jones基本结构具有很好的性质。具有右伴随的双模映射的C∗-代数是紧的∗空间上的有限维C Hausdorff-代数的连续域，其纤维维度由指数有界。如果A是单位数，则右指标属于A当且仅当X是有限生成的右模。有限指标双模是同时具有有限右指标和有限左指标的双Hilbertian C∗-双模。双Hilbertian有限指标C∗-双模，当被认为是右Hilbertian C∗-双模的张量2-C∗-范畴的对象时，在Longo和Roberts意义下，恰好是那些在同一范畴中具有共轭的对象。
We introduce the notion of finite right (or left) numerical index on a C∗-bimoduleAXBwith a bi-Hilbertian structure, based on a Pimsner–Popa-type inequality. The right index of X can be constructed in the centre of the enveloping von Neumann algebra of A . The bimodule X is called of finite right index if the right index lies in the multiplier algebra of A. In this case the Jones basic construction enjoys nice properties. The C∗-algebra of bimodule mappings with a right adjoint is a continuous field of finite dimensional C∗-algebras over a compact Hausdorff space, whose fiber dimensions are bounded above by the index. If A is unital, the right index belongs to A if and only if X is finitely generated as a right module. A finite index bimodule is a bi-Hilbertian C∗-bimodule which is at the same time of finite right and left index. Bi-Hilbertian, finite index C∗-bimodules, when regarded as objects of the tensor 2-C∗-category of right Hilbertian C∗-bimodules, are precisely those objects with a conjugate in the same category, in the sense of Longo and Roberts.