KMS states of a generalized Toeplitz algebras

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 $( \mathcal{T} ( \mathrm{P}\rtimes\Bbb N^{\times}), \Bbb R, \sigma ).$