Commutative C*-Algebras of Toeplitz Operators on the Unit Ball, I. Bargmann-Type Transforms and Spectral Representations of Toeplitz Operators (original) (raw)

C*-algebra of angular Toeplitz operators on Bergman spaces over the upper half-plane

Communications in Mathematical Analysis, 2014

We consider the C*-algebra generated by Toeplitz operators acting on the Bergman space over the upper half-plane whose symbols depend only on the argument of the variable. This algebra is known to be commutative, and it is isometrically isomorphic to a certain algebra of bounded complex-valued functions on the real numbers. In the paper we prove that the latter algebra consists of all bounded functions f that are very slowly oscillating on the real line in the sense that the composition of f with sinh is uniformly continuous with respect to the usual metric.

Commutative C*-algebras of Toeplitz operators and quantization on the unit disk

Reporte interno - CINVESTAV, 2003

A family of recently discovered commutative C *-algebras of Toeplitz operators on the unit disk can be classified as follows. Each pencil of hyperbolic straight lines determines a set of symbols consisting of functions which are constant on the corresponding cycles, the orthogonal trajectories to lines forming a pencil. The C *-algebra generated by Toeplitz operators with such symbols turns out to be commutative. We show that these cases are the only possible ones which generate the commutative C *-algebras of Toeplitz operators on each weighted Bergman space.

Algebras of Toeplitz Operators on the n-Dimensional Unit Ball

Complex Analysis and Operator Theory

We study C *-algebras generated by Toeplitz operators acting on the standard weighted Bergman space A 2 λ (B n) over the unit ball B n in C n. The symbols f ac of generating operators are assumed to be of a certain product type, see (1.1). By choosing a and c in different function algebras S a and S c over lower dimensional unit balls B ℓ and B n−ℓ , respectively, and by assuming the invariance of a ∈ S a under some torus action we obtain C *-algebras T λ (S a , S c) whose structural properties can be described. In the case of k-quasi-radial functions S a and bounded uniformly continuous or vanishing oscillation symbols S c we describe the structure of elements from the algebra * This work was partially supported by CONACYT Project 238630, México and by DFG (Deutsche Forschungsgemeinschaft), Project BA 3793/4-1. T λ (S a , S c), derive a list of irreducible representations of T λ (S a , S c), and prove completeness of this list in some cases. Some of these representations originate from a "quantization effect", induced by the representation of A 2 λ (B n) as the direct sum of Bergman spaces over a lower dimensional unit ball with growing weight parameter. As an application we derive the essential spectrum and index formulas for matrix-valued operators.

C∗-algebra generated by angular Toeplitz operators on the weighted Bergman spaces over the upper half-plane

2016

We consider the set of all Toeplitz operators acting on the weighted Bergman space over the upper half-plane whose L ∞-symbols depend only on the argument of the polar coordinates. The main result states that the uniform closure of this set coincides with the C *-algebra generated by the above Toeplitz operators and is isometrically isomorphic to the C *-algebra of bounded functions that are very slowly oscillating on the real line in the sense that they are uniformly continuous with respect to the arcsinh-metric on the real line.

Toeplitz Operators on the Weighted Bergman Space over the Two-Dimensional Unit Ball

Journal of Function Spaces, 2015

We extend the known results on commutative Banach algebras generated by Toeplitz operators with radial quasi-homogeneous symbols on the two-dimensional unit ball. Spherical coordinates previously used hid a possibility to detect an essentially wider class of symbols that can generate commutative Banach Toeplitz operator algebras. We characterize these new algebras describing their properties and, under a certain extra condition, construct the corresponding Gelfand theory.

Commutative -algebras of Toeplitz operators and quantization on the unit disk

Journal of Functional Analysis, 2006

A family of recently discovered commutative C * -algebras of Toeplitz operators on the unit disk can be classified as follows. Each pencil of hyperbolic straight lines determines a set of symbols consisting of functions which are constant on the corresponding cycles, the orthogonal trajectories to lines forming a pencil. The C * -algebra generated by Toeplitz operators with such symbols turns out to be commutative. We show that these cases are the only possible ones which generate the commutative C * -algebras of Toeplitz operators on each weighted Bergman space.  2005 Elsevier Inc. All rights reserved.

On algebras generated by Toeplitz operators and their representations

Journal of Functional Analysis, 2017

We study Banach and C *-algebras generated by Toeplitz operators acting on weighted Bergman spaces A 2 λ (B 2) over the complex unit ball B 2 ⊂ C 2. Our key point is an orthogonal decomposition of A 2 λ (B 2) into a countable sum of infinite dimensional spaces, each one of which can be identified with a differently weighted Bergman space A 2 μ (D) over the complex unit disk D. Moreover, all elements of the above algebras leave each of the summands in the above decomposition invariant and their restriction to each level acts as a compact perturbation of a Toeplitz operator on A 2 μ (D). The symbols of the generating Toeplitz operators are chosen to be suitable extensions to B 2 of families S of bounded functions on D. Symbol classes S that generate important classical commutative and non-commutative Toeplitz algebras in L(A 2 μ (D)) are of particular interest. In this paper we discuss various examples. In the case of S = C(D) and S = C(D) ⊗ L ∞ (0, 1) we characterize all irreducible representations of the resulting Toeplitz operator C *-algebras. Their Calkin algebras are described and * This work was partially supported by CONACYT Project 238630, México and through the Deutsche Forschungsgemeinschaft, DFG Sachmittelbeihilfe BA 3793/4-1. 1 index formulas are provided.