On the index of product systems of Hilbert modules

TL;DR

This paper introduces a new way to define and analyze the index of product systems of Hilbert modules over C*-algebras, generalizing previous concepts and establishing its mathematical properties.

Contribution

It constructs a Hilbert module structure on the set of units, defines a generalized index, and proves its functoriality and subadditivity properties.

Findings

The index forms a covariant functor from product systems to bimodules.

The index is subadditive under tensor products.

The index has specific properties for embedded spatial product systems.

Abstract

In this note we prove that the set of all uniformly continuous units on a product system over a C* algebra B can be endowed with the structure of left right 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 the generalization of earlier defined indices by Arveson (in the case B=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 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.