Convergence of Tsirelson convolution systems of probability spaces

TL;DR

This paper investigates the convergence properties of Tsirelson convolution systems of probability spaces, establishing links to continuous product systems, flow systems, and two-parameter Hilbert space systems.

Contribution

It introduces the concepts of convergence and K-convergence for Tsirelson convolution systems and explores their implications for continuous and flow systems.

Findings

Convergent systems lead to continuous products of probability spaces.

K-convergent systems give rise to flow systems.

Relationships between convergence, K-convergence, and Hilbert space systems are analyzed.

Abstract

We associate two specific projective systems of probability spaces with any Tsirelson convolution system. If the projective limits of these systems exist, then we call the convolution system convergent and $K$-convergent, respectively. It is shown that convergent convolution systems give rise to continuous products of probability spaces, while $K$-convergent convolution systems lead to flow systems. We investigate the relationship between convergence and $K$-convergence, as well as their connections to two-parameter product systems of Hilbert spaces.