Quotient Algebras of Toeplitz-Composition $$C^{*}$$ C ∗ -Algebras for Finite Blaschke Products

Abstract

Let R be a finite Blaschke product. We study the C * -algebra T C R generated by both the composition operator C R and the Toeplitz operator Tz on the Hardy space. We show that the simplicity of the quotient algebra OC R by the ideal of the compact operators can be characterized by the dynamics near the Denjoy-Wolff point of R if the degree of R is at least two. Moreover we prove that the degree of finite Blaschke products is a complete isomorphism invariant for the class of OC R such that R is a finite Blaschke product of degree at least two and the Julia set of R is the unit circle, using the Kirchberg-Phillips classification theorem.

$C^*$-algebras generated by multiplication operators and composition operators with rational symbol

“…Watatani and the author [5] proved that the quotient algebra OC R is isomorphic to the C * -algebra O R (J R ) associated with the complex dynamical system introduced in [11]. In [4] we extend this result for general finite Blaschke products. Let R be a finite Blaschke product R of degree at least two.…”

“…Let R be a finite Blaschke product of degree at least two. In [3], we proved that there is a relation between a C * -algbara generated by a composition operator C R and Toeplitz operators and the C * -algebra O R (J R ) associated with the complex dynamical system introduced in [12].…”

“…Several authors considered C * -algebras generated by composition operators (and Toeplitz operators) on the Hardy space H 2 (D) on the open unit disk D ( [2,5,7,8,11,12,13,15,17,18,19]). On the other hand, there are some studies on C * -algebras generated by composition operators on L 2 spaces, for example [3,4,14].…”

C*-algebras generated by multiplication operators and composition operators by functions with self-similar branches II

“…Several authors considered C * -algebras generated by composition operators (and Toeplitz operators) on the Hardy space H 2 (D) on the open unit disk D ( [2,5,7,8,14,15,16,18,20,21,22]). On the other hand, there are some studies on C *algebras generated by composition operators on L 2 spaces, for example [3,4,17].…”

Convolution of two harmonic mappings in the right-half plane

“…where c ∈ ∂D = {z : |z| = 1}, m is a nonnegative integers, then we say that B(z) is a finite Blaschke product of degree m. Blaschke products have been applied to many research fields, for instance, Kraus and Gorkin [1], Dallakyan [6], Hamada [7], Akeroyd and Roth [9].…”