On the index of product systems of Hilbert modules

Abstract

In this note we prove that the set of all uniformly continuous units on a product system over a
$C^*$ algebra $\mathcal B$ can be endowed with the structure of left right $\mathcal B$ - $\mathcal B$
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 the generalization of earlier defined
indices by Arveson (in the case $\mathcal B=\mathrm C$) and Skeide (in the case of spatial product system).
We prove that such defined index is a covariant functor from the category od continuous product systems to
the category of $\mathcal B$ 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.