Toeplitz algebras on the disk ✩
نویسندگان
چکیده
Let B be a Douglas algebra and let B be the algebra on the disk generated by the harmonic extensions of the functions in B. In this paper we show that B is generated by H∞(D) and the complex conjugates of the harmonic extensions of the interpolating Blaschke products invertible in B. Every element S in the Toeplitz algebra TB generated by Toeplitz operators (on the Bergman space) with symbols in B has a canonical decomposition S = T S̃ + R for some R in the commutator ideal CTB ; and S is in CTB iff the Berezin transform S̃ vanishes identically on the union of the maximal ideal space of the Douglas algebra B and the set M1 of trivial Gleason parts. © 2006 Elsevier Inc. All rights reserved.
منابع مشابه
Commutative C ∗ - algebras of Toeplitz operators and quantization on the unit disk ✩
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 t...
متن کاملCommutative C ∗ - algebras of Toeplitz operators on the unit ball , II . Geometry of the level sets of symbols
In the first part [16] of this work, we described the commutative C∗algebras generated by Toeplitz operators on the unit ball B whose symbols are invariant under the action of certain Abelian groups of biholomorphisms of B. Now we study the geometric properties of these symbols. This allows us to prove that the behavior observed in the case of the unit disk (see [3]) admits a natural generaliza...
متن کاملAlgebras of Toeplitz operators on the unit ball
One of the common strategies in the study of Toeplitz operators consists in selecting of various special symbol classes S ⊂ L∞ so that the properties of both the individual Toeplitz operators Ta, with a ∈ S, and of the algebra generated by such Toeplitz operators can be characterized. A motivation to study an algebra generated by Toeplitz operators (rather than just Toeplitz operators themselve...
متن کاملQuasi-radial quasi-homogeneous symbols and commutative Banach algebras of Toeplitz operators
We present here a quite unexpected result: Apart from already known commutative C∗-algebras generated by Toeplitz operators on the unit ball, there are many other Banach algebras generated by Toeplitz operators which are commutative on each weighted Bergman space. These last algebras are non conjugated via biholomorphisms of the unit ball, non of them is a C∗-algebra, and for n = 1 all of them ...
متن کاملParabolic quasi-radial quasi-homogeneous symbols and commutative algebras of Toeplitz operators∗
We describe new Banach (not C∗ !) algebras generated by Toeplitz operators which are commutative on each weighted Bergman space over the unit ball B, where n > 2. For n = 2 all these algebras collapse to the single C∗-algebra generated by Toeplitz operators with quasi-parabolic symbols. As a by-product, we describe the situations when the product of mutually commuting Toeplitz operators is a To...
متن کامل