Invariant states on noncommutative tori

TL;DR

This paper proves that for noncommutative tori with irrational deformation parameter, the only invariant state under the symplectic automorphisms is the canonical trace, highlighting a unique symmetry property.

Contribution

It establishes the uniqueness of the invariant state on noncommutative tori under symplectic automorphisms for irrational parameters.

Findings

The trace state is the only invariant state under symplectic automorphisms.

Invariant states are uniquely determined by the trace in this setting.

The result applies to noncommutative tori with irrational deformation parameters.

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 $\tau$.