KMS states of a generalized Toeplitz algebras

TL;DR

This paper investigates the structure of KMS states for a generalized Toeplitz algebra associated with a non-quasi-lattice ordered semigroup, providing explicit calculations for the states in a specific dynamical system.

Contribution

It introduces and analyzes a generalized Toeplitz algebra for a non-quasi-lattice semigroup and computes the KMS states for its natural $C^*$-dynamical system.

Findings

Explicit KMS state values computed for the system.

Extension of Toeplitz algebra framework to non-quasi-lattice semigroups.

Characterization of the algebra's structure in this context.

Abstract

In this paper, we consider a generalized Toeplitz algebra $\mathcal{T} ( \mathrm{P}\rtimes\Bbb N^{\times})$ for a non-quasi-lattice ordered semigroup $ \mathrm{P}\rtimes\Bbb N^{\times}$ where $ \mathrm{P}\rtimes\Bbb N^{\times}$ is a semidirect product of an additive semigroup $ \mathrm{P} = \{0, 2, 3, \cdots \}$ by a multiplicative positive natural numbers semigroup $ \Bbb N^{\times}$. And also we compute the values of the KMS state of the natural $C^*$-dynamical system $( \mathcal{T} ( \mathrm{P}\rtimes\Bbb N^{\times}), \Bbb R, \sigma ).$