On the Index of Product Systems of Hilbert Modules

Dragoljub J. Kečkić, Biljana Vujošević

In this note we prove that the set of all uniformly continuous units on a product system over a $C^*$ algebra $\cal B$ can be endowed with a structure of left-right $\cal{B-B}$ Hilbert module after identifying similar units by the suitable equivalence relation. We use this construction to define the index of the initial product system, and prove that it is a generalization of earlier defined indices by Arveson (in the case ${\cal B}=C$) and Skeide (in the case of spatial product system). We prove that such defined index is a covariant functor from the category of continuous product systems to the category of $\cal B$ bimodules. We also prove that the index is subadditive with respect to the outer tensor product of product systems, and prove additional properties of the index of product systems that can be embedded into a spatial one.