The $C^*$-algebra $\mathfrak{T}_m$ as a crossed product

In this paper we consider the $C^*$-subalgebra $\mathfrak{T}_m$ of the Toeplitz algebra $\mathfrak{T}$ generated by monomials, which have an index divisible by $m$. We present the algebra $\mathfrak{T}_m$ as a crossed product: $\mathfrak{T}_m=\varphi(A)\times_{\delta_m}\mathbb{Z}$, where $A=C_0 (\mathbb{Z}_+)\oplus\mathbb{C}I$ 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 $\mathfrak{T}=\varphi(A)\times_{\delta_1}\mathbb{Z}$, which is an analogue of Coburn’s theorem.