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 $.

On the Classification of Rational Quantum Tori and the Structure of Their Automorphism Groups

An n-dimensional quantum torus is a twisted group algebra of the group ℤn. It is called rational if all invertible commutators are roots of unity. In the present note we describe a normal form for rational n-dimensional quantum tori over any field. Moreover, we show that for n = 2 the natural exact sequence describing the automorphism group of the quantum torus splits over any field.