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

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.