Inclusion systems and amalgamated products of product systems

TL;DR

This paper introduces the amalgamated tensor product of product systems, generalizing the spatial tensor product by allowing contractive morphisms, and develops tools for index computation using inclusion systems.

Contribution

It generalizes the spatial tensor product of product systems to include contractive morphisms and introduces inclusion systems for index calculations.

Findings

Amalgamation index adds up for normalized units.

For non-normalized units, the index is one more than the sum.

Inclusion systems are effective tools for index computations.

Abstract

Here we generalize the concept of spatial tensor product, introduced by Skeide, of two product systems via a pair of normalized units. This new notion is called amalgamated tensor product of product systems, and now the amalgamation can be done using a contractive morphism. Index of amalgamation product (when done through units) adds up for normalized units but for non-normalized units, the index is one more than the sum. We define inclusion systems and use it as a tool for index computations. It is expected that this notion will have other uses.