IUMJ

Title: Quantum exchangeable sequences of algebras

Authors: Stephen Curran

Issue: Volume 58 (2009), Issue 3, 1097-1126

Abstract:

We extend the notion of quantum exchangeability, introduced by K\"{o}stler and Speicher in [K. K\"{o}stler and R. Speicher, \emph{A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation}, http://www.arxiv.org/abs/0807.0677 (Preprint)], to sequences $(\rho_1,\rho_2,\dotsc)$ of homomorphisms from an algebra $C$ into a noncommutative probability space $(A,\varphi)$, and prove a free de Finetti theorem in this context: an infinite quantum exchangeable sequence $(\rho_1,\rho_2,\dotsc)$ is freely independent and identically distributed with respect to a conditional expectation. As in the classical case, the theorem fails for finite sequences. We give a characterization of finite quantum exchangeable sequences, which can be viewed as a noncommutative analogue of the classical urn sequences. We then give an approximation to how far a finite quantum exchangeable sequence is from being freely independent with amalgamation.