Invariant states on noncommutative tori

Abstract

For any number $h$ such that $\hbar:=h/2\pi$ is irrational and any skew-symmetric, non-degenerate bilinear form $\sigma:\mathbb{Z}^{2g}\times \mathbb{Z}^{2g} \to \mathbb{Z}$, let be $\mathcal{A}^h_{g,\sigma}$ be the twisted group $*$-algebra $\mathbb{C}[\mathbb{Z}^{2g}]$ and consider the ergodic group of $*$-automorphisms of $\mathcal{A}^h_{g,\sigma}$ induced by the action of the symplectic group Sp$(\mathbb{Z}^{2g},\sigma)$. We show that the only Sp$(\mathbb{Z}^{2g},\sigma)$-invariant state on $\mathcal{A}^h_{g,\sigma}$ is the trace state $\tau$.