Invariant States on Noncommutative Tori

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 $\textrm{Sp} \,({{\mathbb{Z}}}^{2g},\sigma )$. We show that the only $\textrm{Sp} \,({{\mathbb{Z}}}^{2g},\sigma )$-invariant state on ${{\mathcal{A}}}^h_{g,\sigma }$ is the trace state $\tau $.