From random sets to continuous tensor products: answers to three questions of W. Arveson

TL;DR

This paper explores the construction of continuous tensor product systems from stochastic processes, demonstrating how replacing Brownian motion with Bessel processes yields a variety of non-isomorphic product systems, thus addressing fundamental questions in operator algebra theory.

Contribution

It introduces a method to generate a continuum of non-isomorphic product systems by replacing Brownian motion with Bessel processes, answering three key questions posed by W. Arveson.

Findings

Replacing Brownian motion with Bessel processes produces non-isomorphic product systems.

The work provides new insights into the structure of continuous tensor product systems.

It advances understanding of the relationship between stochastic processes and operator algebra structures.

Abstract

The set of zeros of a Brownian motion gives rise to a product system in the sense of William Arveson (that is, a continuous tensor product system of Hilbert spaces). Replacing the Brownian motion with a Bessel process we get a continuum of non-isomorphic product systems.