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

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.