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

Abstract

We introduce the notion of finite right (respectively left) numerical index on a bimodule $X$ over C*-algebras A and B with a bi-Hilbertian structure. This notion is based on a Pimsner-Popa type inequality. The right (respectively left) index element of X can be constructed in the centre of the enveloping von Neumann algebra of A (respectively B). X is called of finite right index if the right index element lies in the multiplier algebra of A. In this case we can perform the Jones basic construction. Furthermore the C*--algebra of bimodule mappings with a right adjoint is a continuous field of finite dimensional C*-algebras over the spectrum of Z(M(A)), whose fiber dimensions are bounded above by the index. We show that if A is unital, the right index element belongs to A if and only if X is finitely generated as a right module. We show that bi-Hilbertian, finite (right and left) index C*-bimodules are precisely those objects of the tensor 2-C*-category of right Hilbertian C*-bimodules with a conjugate object, in the sense of Longo and Roberts, in the same category.