THE $ C^*$-ALGEBRA $\mathcal{T}_m$ AS A CROSSED PRODUCT - Hovsepyan, K.H. (Proceedings of the YSU A: Physical and Mathematical Sciences, 2014)
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: $\mathcal{T}_m=φ(\mathcal{A})×_{δ_m}\mathbb{Z}$, 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 $\mathcal{T}=φ(\mathcal{A})×_{δ_1}\mathbb{Z}$, which is an analogue of Coburn's theorem.
Keywords: index of monomial, coefficient algebra, crossed product, finitely representable, Toeplitz algebra, $C^*$-algebra, transfer operator