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

By Tsuyoshi Kajiwara, C. Pinzari, Y. Watatani. 2003

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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.

Subfields: Algebra and Number Theory, Applied Mathematics, Mathematical Physics

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