Semicrossed Products of the Disk Algebra and the Jacobson Radical

Abstract

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 positive hyperbolic step, the Jacobson radical is nonzero and a proper subset of the set of quasinilpotent elements.

Unicritical Blaschke Products and Domains of Ellipticity

“…We first give the case where the endpoints are never fixed. 8 Lemma 3.6. Suppose that (n − 1)ψ ∈ {2kπ : k ∈ Z} if n is even or (n − 1)ψ ∈ {(2k + 1)π : k ∈ Z} if n is odd.…”

“…For finite Blaschke products, the Julia set is always contained in ∂D and is either the whole of ∂D or a Cantor subset of ∂D. These two cases can be characterized as follows; see [3, p.58] and [1,4] as well as [8] for a discussion of this characterization. Theorem 1.1.…”

“…In this case, B is said to be of positive hyperbolic step. See [4], as well as [8], for more details. 1.5.…”