THE $ C^*$-ALGEBRA $\mathcal{T}_m$ AS A CROSSED PRODUCT

Abstract

In this paper we consider the $ C^*$-subalgebra $\mathcal{T}_m$ of the Toeplitz algebra  $\mathcal{T}$ generated by monomials, which have an index divisible by $m$. We present the algebra $\mathcal{T}_m$ as a crossed product: $\mathcal{T}_m=\varphi(\mathcal{A})\times_{\delta_m}\mathbb{Z}$, where $\mathcal{A}=C_0(\mathbb{Z}_+)\oplus \mathbb{C}I$ is $ C^*$-algebra of all continuous functions on $\mathbb{Z}_+$, which have a finite limit at infinity. In the case $m=1$ we obtain that $\mathcal{T}=\varphi(\mathcal{A})\times_{\delta_1}\mathbb{Z}$, which is an analogue of Coburn's theorem.