Toeplitz-composition $C^{*}$-algebras for certain finite Blaschke products

Abstract

Abstract. Let R be a finite Blaschke product of degree at least two with R(0) = 0. Then there exists a relation between the associated composition operator C R on the Hardy space and the C * -algebra O R (J R ) associated with the complex dynamical system (R •n ) n on the Julia set J R . We study the C * -algebra T C R generated by both the composition operator C R and the Toeplitz operator T z to show that the quotient algebra by the ideal of the compact operators is isomorphic to the C * -algebra O R (J R ), which is simple and purely infinite.

Composition Operators and Endomorphisms

“…We call this isometry the master isometry determined by b (or by the endomorphism β induced by b. ) Much of the material below is contained in results already in the literature (see in particular [8,16]). But many calculations are done under the additional hypothesis that b(0) = 0, which we want specifically to avoid.…”

“…Thus there are crossed-product-like structures in the background of our analysis. We have not directly exploited this structure in our work as it was not necessary to do so, but it has been used, for example, by Hamada and Watatani [8] in their study of C * -algebras associated with the operators b , and we expect that it will play a key role in future work connecting composition operators with operator algebras. See [7] for further information about Cuntz-Pimsner algebras, their representations and their connections with crossed products.…”

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

“…Let R be a finite Blaschke product of degree at least two with R(0) = 0. Watatani and the author [10] 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 [15]. The C * -algebra O R (J R ) is defined as a Cuntz-Pimsner algebra [28].…”

“…In the proof of this theorem, one of the keys is to analyze operators of the form C * R T a C R for a ∈ L ∞ (T). Watatani and the author [10] showed that, if R is a finite Blaschke product of degree at least two with R(0) = 0, then the operator C * R T a C R is a Toeplitz operator T LR(a) . Courtney, Muhly and Schmidt [5] extend this to the case for a general finite Blaschke product.…”

Endomorphisms, composition operators and Cuntz families

“…We refer to C b as the master isometry determined by b (or by the endomorphism β induced by b.) Much of the material below is contained in results already in the literature (see in particular [16] and [7]). But many calculations are done under the additional hypothesis that b(0) = 0, which we want specifically to avoid.…”