Tracial States on the $\theta$-deformed Plane

Abstract

Starting from a trivial pure state $\Psi^{0}$, we construct non-trivial tracial states $(\Psi^{i})$ on the $\theta$-deformed $2m$-plane $C^{alg}(\mathbb{R}_{\theta}^{2m})$. Furthermore we generalize $\Psi^{i}$ to another tracial state on $C^{alg}(\mathbb{R}_{\theta}^{2m})$. We study extreme points of the tracial state space of $C^{alg}(\mathbb{R}_{\theta}^{2m})$ in the case that deformation parameters are irrational numbers. Non-trivial pure states $(\Phi_{t}^{k})$ on $C^{alg}(\mathbb{R}_{\theta}^{2m})$ are also given.