Noncommutative continuous Bernoulli shifts

TL;DR

This paper extends classical stochastic noise concepts into the non-commutative setting, introducing continuous Bernoulli shifts and establishing their structure, connections to operator algebras, and applications to quantum probability and free probability.

Contribution

It introduces non-commutative continuous Bernoulli shifts, explores their structure, and establishes a bijective correspondence between additive and unital shift cocycles.

Findings

Von Neumann algebra of a continuous Bernoulli shift is either finite or type III.

Cocycles of these shifts serve as an operator algebraic notion for Levy processes.

Examples are provided from probability, quantum probability, and free probability.

Abstract

We introduce a non-commutative extension of Tsirelson-Vershik's noises, called (non-commutative) continuous Bernoulli shifts. These shifts encode stochastic independence in terms of commuting squares, as they are familiar in subfactor theory. Such shifts are, in particular, capable of producing Arveson's product system of type I and type II. We investigate the structure of these shifts and prove that the von Neumann algebra of a (scalar-expected) continuous Bernoulli shift is either finite or of type III. The role of (`classical') stationary flows for Tsirelson-Vershik's noises is now played by cocycles of continuous Bernoulli shifts. We show that these cocycles provide an operator algebraic notion for Levy processes. They lead, in particular, to units and `logarithms' of units in Arveson's product systems. Furthermore, we introduce (non-commutative) white noises, which are operator algebraic versions of Tsirelson's `classical' noises. We give examples coming from probability, quantum probability and from Voiculescu's theory of free probability. Our main result is a bijective correspondence between additive and unital shift cocycles. For the proof of the correspondence we develop tools which are of interest on their own: non-commutative extensions of stochastic Ito integration, stochastic logarithms and exponentials.