KMS states on $C_c^{\ast }({\mathbb{N}}^2)$

Abstract

Let $C_c^{\ast }({\mathbb{N}}^2)$ be the universal $C^{\ast }$-algebra generated by a semigroup of isometries $\{v_{(m,n)}:m,n\in \mathbb{N}\}$ whose range projections commute. We analyse the structure of KMS states on $C_c^{\ast }({\mathbb{N}}^2)$ for the time evolution determined by a homomorphism $c:{\mathbb{Z}}^2\to \mathbb{R}$. In contrast to the reduced version $C_{red}^{\ast }({\mathbb{N}}^2)$, we show that the set of KMS states on $C_c^{\ast }({\mathbb{N}}^2)$ has a rich structure. In particular, we exhibit uncountably many extremal KMS states of type I, II and III.