TY - JOUR

T1 - Jones index theory for Hilbert C*-bimodules and its equivalence with conjugation theory

AU - Kajiwara, Tsuyoshi

AU - Pinzari, Claudia

AU - Watatani, Yasuo

N1 - Funding Information:
·Corresponding author. Fax: +39-06-44701007. E-mail address: pinzari@mat.uniroma1.it (C. Pinzari). 1Partially supported by Grants-in-Aid for Scientific Research 12640210 and 14340050 from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

PY - 2004/10/1

Y1 - 2004/10/1

N2 - We introduce the notion of finite right (or left) numerical index on a C*-bimodule AXB with 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.

AB - We introduce the notion of finite right (or left) numerical index on a C*-bimodule AXB with 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.

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U2 - 10.1016/j.jfa.2003.09.008

DO - 10.1016/j.jfa.2003.09.008

M3 - Article

AN - SCOPUS:4344592567

VL - 215

SP - 1

EP - 49

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 1

ER -