De Finetti theorem on the infinte non-commutative torus

The set of spreadable states on an infinite non-commutative torus $\mathbb{A}_\a^\bz$ is determined for all values of the deformation parameter $\a$. If $\frac{\a}{2\pi}$ is irrational, the canonical trace is the only spreadable state. If $\frac{\a}{2\pi}$ is rational, the set of all spreadable states is a Bauer simplex. Moreover, its boundary is the set of all infinite products of a single state on $C(\bt)$, which is invariant under all rotations by $n_0$-th roots of unity, where $n_0=p_1^{\{\frac{m_1}{2}\}}\cdots p_r^{\{\frac{m_r}{2}\}}$, with $\{\frac{n}{2}\}$ being $\frac{n}{2}$ for $n$ even and $\frac{n+1}{2}$ for $n$ odd, if $\frac{q_1^{n_1}\ldots q_s^{n_s}}{p_1^{m_1}\ldots p_r^{m_r}}$ is the representation of $\frac{\a}{2\pi}$ in lowest terms.\\ Finally, the simplex of all stationary states on $\mathbb{A}_\a^\bz$ is proved to be the Poulsen simplex for all values of the deformation parameter $\a$.