Invariant states on noncommutative tori

Abstract

For any number $h$ such that $\hslash :=h/2\pi$ is irrational and any skew-symmetric, non-degenerate bilinear form $\sigma :{\mathbb{Z}}^{2g}\times {\mathbb{Z}}^{2g}\to \mathbb{Z}$, let be ${\mathcal{A}}_{g,\sigma }^h$ be the twisted group $\ast$-algebra $\mathbb{C}[{\mathbb{Z}}^{2g}]$ and consider the ergodic group of $\ast$-automorphisms of ${\mathcal{A}}_{g,\sigma }^h$ 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 ${\mathcal{A}}_{g,\sigma }^h$ is the trace state $\tau$.

11 pages - accepted in International Mathematics Research Notices

References in corpus (1)

• Non-existence of natural states for Abelian Chern-Simons theory

Cited by in corpus (3)

• Intertwining operators for symmetric hyperbolic systems on globally hyperbolic manifolds
• Injective tensor products in strict deformation quantization
• On the uniqueness of invariant states