The Relation of Spatial and Tensor Product of Arveson Systems --- The Random Set Point of View

TL;DR

This paper characterizes the embedding of the spatial product of two Arveson systems into their tensor product using random set techniques, showing the independence of the spatial tensor product from reference units and analyzing cases where they differ.

Contribution

It introduces a random set approach to understand the embedding of spatial products into tensor products of Arveson systems, establishing their intrinsic nature and exploring examples from Bessel process zeros.

Findings

Spatial tensor product is independent of reference units.

Examples from Bessel zeros show non-coincidence of products.

Bessel-derived Arveson systems are primitive.

Abstract

We characterise the embedding of the spatial product of two Arveson systems into their tensor product using the random set technique. An important implication is that the spatial tensor product does not depend on the choice of the reference units, i.e. it is an intrinsic construction. There is a continuous range of examples coming from the zero sets of Bessel processes where the two products do not coincide. The lattice of all subsystems of the tensor product is analised in different cases. As a by-product, the Arveson systems coming from Bessel zeros prove to be primitive in the sense of \cite{JMP11a}.