On Lehner's `free' noncommutative analogue of De Finetti's theorem

TL;DR

This paper introduces and proves the equivalence of two notions, 'weak conditional freeness' and 'conditional freeness', for stationary processes in noncommutative probability, extending Lehner's work and embedding these processes into an operator algebraic framework.

Contribution

It defines new concepts of freeness in noncommutative probability and establishes their equivalence, linking them to von Neumann algebraic amalgamated free products.

Findings

'Weak conditional freeness' and 'conditional freeness' are equivalent.

Stationary processes with these properties embed into von Neumann algebraic amalgamated free products.

The work extends Lehner's results to a broader noncommutative setting.

Abstract

Inspired by Lehner's results on exchangeability systems we define `weak conditional freeness' and `conditional freeness' for stationary processes in an operator algebraic framework of noncommutative probability. We show that these two properties are equivalent and thus the process embeds into a von Neumann algebraic amalgamated free product over the fixed point algebra of the stationary process.