On cluster systems of tensor product systems of Hilbert spaces

TL;DR

This paper investigates the structure of subsystems within tensor products of Hilbert space product systems, establishing new relations with cluster systems and showing independence of certain amalgamated products from unit choices.

Contribution

It extends known results on the spatial product of product systems by analyzing subsystems and their relation to cluster systems, including a case of unit-independent amalgamated products.

Findings

Amalgamated product through strictly contractive units is independent of unit choices.

The amalgamated product is isomorphic to the tensor product of the spatial product and a type I system.

Relations between subsystems and cluster systems are established.

Abstract

It is known that the spatial product of two product systems is intrinsic. Here we extend this result by analyzing subsystems of the tensor product of product systems. A relation with cluster systems is established. In a special case, we show that the amalgamated product of product systems through strictly contractive units is independent of the choices of the units. The amalgamated product in this case is isomorphic to the tensor product of the spatial product of the two and the type I product system of index one.