De Finetti Theorem on the infinite non-commutative torus

The set of spreadabl estates on an infinite non-commutive torus $\mathbb{A}$_{$\mathbb{Z}$α} is determined for all values of the deformation parameter α. If α is irrational, the canonical trace is the only spreadable 2π state. If α is rational, the set of all spreadable states is a Bauer 2π 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 $\mathbb{A}${$\mathbb{Z}$_α} is proved to be the Poulsen simplex for all values of the deformation parameter α.