### Abstract

We introduce the notion of finite right (or left) numerical index on a C^{*}-bimodule _{A}X_{B} 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.

Original language | English |
---|---|

Pages (from-to) | 1-49 |

Number of pages | 49 |

Journal | Journal of Functional Analysis |

Volume | 215 |

Issue number | 1 |

DOIs | |

Publication status | Published - Oct 1 2004 |

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### All Science Journal Classification (ASJC) codes

- Analysis

### Cite this

^{*}-bimodules and its equivalence with conjugation theory.

*Journal of Functional Analysis*,

*215*(1), 1-49. https://doi.org/10.1016/j.jfa.2003.09.008

**Jones index theory for Hilbert C ^{*}-bimodules and its equivalence with conjugation theory.** / Kajiwara, Tsuyoshi; Pinzari, Claudia; Watatani, Yasuo.

Research output: Contribution to journal › Article

^{*}-bimodules and its equivalence with conjugation theory',

*Journal of Functional Analysis*, vol. 215, no. 1, pp. 1-49. https://doi.org/10.1016/j.jfa.2003.09.008

^{*}-bimodules and its equivalence with conjugation theory. Journal of Functional Analysis. 2004 Oct 1;215(1):1-49. https://doi.org/10.1016/j.jfa.2003.09.008

}

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

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.

UR - http://www.scopus.com/inward/record.url?scp=4344592567&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=4344592567&partnerID=8YFLogxK

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 -