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

Tsuyoshi Kajiwara, Claudia Pinzari, Yasuo Watatani

Research output: Contribution to journalArticle

26 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1-49
Number of pages49
JournalJournal of Functional Analysis
Volume215
Issue number1
DOIs
Publication statusPublished - Oct 1 2004

All Science Journal Classification (ASJC) codes

  • Analysis

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