Endomorphisms of the Toeplitz algebra

This article describes all injective endomorphisms of the classical Toeplitz algebra. Their connection with endomorphisms of the algebra of continuous functions on the unit circle and with coverings over the unit circle was considered. It was shown that each non-unitary isometry $V$ in the Toeplitz algebra determines the identity preserving endomorphism, as well as the class of its compact perturbations, i.e., identity non-preserving endomorphisms, defined by partial isometries $\{VP\}$, where $P$ is a projection of finite codimension. The notions of $\mathcal{T}$-equivalence of endomorphisms and $\mathcal{T}$-equivalence up to a compact perturbation were introduced. An example was provided wherein the isometries are unitarily equivalent but the corresponding endomorphisms fall into different equivalence classes. Of all endomorphisms, the ones belonging to the class of Blaschke endomorphisms, which are analogous to endomorphisms of the disc-algebra and generate unbranched coverings over the unit circle, were singled out.