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

TL;DR

This paper investigates the structure of KMS states on a universal C*-algebra generated by isometries over a5a5^2, revealing a rich set of extremal states of various types, contrasting with the reduced version.

Contribution

It provides a detailed analysis of KMS states on the universal algebra, showing the existence of uncountably many extremal states of types I, II, and III, which was not known before.

Findings

Uncountably many extremal KMS states of type I, II, and III identified.

The structure of KMS states on the universal algebra is richer than on the reduced algebra.

Contrast established between the universal and reduced C*-algebras regarding KMS states.

Abstract

Let $C_c^{*}(\mathbb{N}^{2})$ be the universal $C^{*}$-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 $C_{c}^{*}(\mathbb{N}^2)$ for the time evolution determined by a homomorphism $c:\mathbb{Z}^{2} \to \mathbb{R}$. In contrast to the reduced version $C_{red}^{*}(\mathbb{N}^{2})$, we show that the set of KMS states on $C_{c}^{*}(\mathbb{N}^{2})$ has a rich structure. In particular, we exhibit uncountably many extremal KMS states of type I, II and III.