Tail algebras of quantum exchangeable random variables

TL;DR

This paper demonstrates that any countably generated von Neumann algebra with a normal faithful state can serve as the tail algebra of a quantum exchangeable sequence, and characterizes when the state is a limit of convex combinations of free product states.

Contribution

It establishes a universal construction for tail algebras in quantum exchangeable sequences and characterizes specific state limits, advancing understanding of noncommutative probability structures.

Findings

Any countably generated von Neumann algebra can be realized as a tail algebra.

Characterization of states as limits of convex combinations of free product states.

Provides a framework connecting tail algebras with quantum exchangeability.

Abstract

We show that any countably generated von Neumann algebra with specified normal faithful state can arise as the tail algebra of a quantum exchangeable sequence of noncommutative random variables. We also characterize the cases when the state corresponds to a limit of convex combinations of free products states.