### 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 |
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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 |

### All Science Journal Classification (ASJC) codes

- Analysis

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## 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