Semicrossed products of the disk algebra and the Jacobson radical

TL;DR

This paper studies semicrossed products of the disk algebra under finite Blaschke product endomorphisms, characterizing their Jacobson radical and analyzing the presence of quasinilpotent elements depending on the type of Blaschke product.

Contribution

It provides a detailed characterization of the Jacobson radical for these operator algebras and distinguishes cases based on the nature of the Blaschke product.

Findings

Elliptic Blaschke products lead to radical-free semicrossed products.

Hyperbolic or parabolic Blaschke products with zero hyperbolic step have nonzero Jacobson radical.

The radical is a proper subset of quasinilpotent elements in certain cases.

Abstract

We consider semicrossed products of the disk algebra with respect to endomorphisms defined by finite Blaschke products. We characterize the Jacobson radical of these operator algebras. Furthermore, in the case the finite Blaschke product is elliptic, we show that the semicrossed product contains no nonzero quasinilpotent elements. However, if the finite Blaschke product is hyperbolic or parabolic with zero hyperbolic step, the Jacobson radical is nonzero and a proper subset of the set of quasinilpotent elements.