De Finetti Theorem on the infinite non-commutative torus

TL;DR

This paper characterizes spreadable and stationary states on the infinite non-commutative torus, revealing a unique trace for irrational parameters and a complex simplex structure for rational ones, advancing understanding of non-commutative probability.

Contribution

It provides a complete classification of spreadable states on the infinite non-commutative torus for all deformation parameters, including the structure of stationary states.

Findings

Unique trace for irrational 

Spreadable states form a Bauer 2b0 simplex when  is rational

Stationary states form the Poulsen simplex for all 

Abstract

The set of spreadabl estates on an infinite non-commutive torus \mathbb{A}_{\mathbb{Z}_\alpha} is determined for all values of the deformation parameter {\alpha}. If {\alpha} is irrational, the canonical trace is the only spreadable 2{\pi} state. If {\alpha} is rational, the set of all spreadable states is a Bauer 2{\pi} simplex. Moreover, its boundary is the set of all infinite products of a single state on C(T). Finally, the simplex of all stationary states on \mathbb{A}_{\mathbb{Z}_\alpha} is proved to be the Poulsen simplex for all values of the deformation parameter {\alpha}.