Quasi-Radial Quasi-Homogeneous Symbols and Commutative Banach Algebras of Toeplitz Operators

Vasilevski, Nikolai

doi:10.1007/s00020-009-1732-8

Cited by 40 publications

(72 citation statements)

References 7 publications

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“…In this paper we aim to analyze operator algebras that are generated by certain families of Toeplitz operators (also see the results in [1,6,7,13,14,15,16,18]). For general symbols a ∈ L ∞ (B ℓ ) and c ∈ L ∞ (B n−ℓ ) not much can be said about the structure of the algebra generated by all Toeplitz operators T λ fac .…”

Section: Corollary 35 Under the Assumptions Of Proposition 34 We Havementioning

confidence: 99%

Algebras of Toeplitz Operators on the n-Dimensional Unit Ball

Bauer

Hagger

Vasilevski

2018

Complex Anal. Oper. Theory

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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.

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Section: Corollary 35 Under the Assumptions Of Proposition 34 We Havementioning

confidence: 99%

Algebras of Toeplitz Operators on the n-Dimensional Unit Ball

Bauer

Hagger

Vasilevski

2018

Complex Anal. Oper. Theory

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“…. , α m ) be a tuple in Z m + such that |α| = α 1 + · · · + α m = k. Similar to [1,6] we divide the coordinates of z ∈ C k into m groups as follows: ,1 , . .…”

Section: Definition 41mentioning

confidence: 99%

“…Recently and quite unexpectedly it was observed in [6] that for n > 1 there are many other, not geometrically defined, classes of symbols which generate commutative Toeplitz operator algebras on each weighted Bergman space. These classes of symbols were in a sense originated from, or subordinated to the quasi-elliptic group, the corresponding commutative operator algebras were Banach, and being extended to C * -algebras they became non-commutative.…”

Section: Introductionmentioning

confidence: 99%

“…These results were extended in [1,7] then to the classes of symbols, subordinated to the quasi-hyperbolic and quasiparabolic groups, which as well generate via corresponding Toeplitz operators classes of Banach algebras being commutative on each weighted Bergman space. That is, together with [6], these papers cover the multidimensional After many unsuccessful attempts to find commutative algebras generated by Toeplitz operators and subordinated to the nilpotent group we conjecture that apart from the known cases there are no more new Banach algebras generated by Toeplitz operators with symbols subordinated to the nilpotent group of biholomorphisms of the unit ball B n and commutative on each weighted Bergman space.…”

Section: Introductionmentioning

confidence: 99%

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Commutative Toeplitz Banach Algebras on the Ball and Quasi-Nilpotent Group Action

Bauer

Vasilevski

2011

Integr. Equ. Oper. Theory

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Studying commutative C * -algebras generated by Toeplitz operators on the unit ball it was proved that, given a maximal commutative subgroup of biholomorphisms of the unit ball, the C * -algebra generated by Toeplitz operators, whose symbols are invariant under the action of this subgroup, is commutative on each standard weighted Bergman space. There are five different pairwise non-conjugate model classes of such subgroups: quasi-elliptic, quasi-parabolic, quasi-hyperbolic, nilpotent, and quasi-nilpotent. Recently it was observed in Vasilevski (Integr Equ Oper Theory. 66:141-152, 2010) that there are many other, not geometrically defined, classes of symbols which generate commutative Toeplitz operator algebras on each weighted Bergman space. These classes of symbols were subordinated to the quasi-elliptic group, the corresponding commutative operator algebras were Banach, and being extended to C * -algebras they became non-commutative. These results were extended then to the classes of symbols, subordinated to the quasihyperbolic and quasi-parabolic groups. In this paper we prove the analogous commutativity result for Toeplitz operators whose symbols are subordinated to the quasi-nilpotent group. At the same time we conjecture that apart from the known C * -algebra cases there are no more new Banach algebras generated by Toeplitz operators whose symbols are subordinated to the nilpotent group and which are commutative on each weighted Bergman space.Mathematics Subject Classification (2010). Primary 47B35; Secondary 47L80, 32A36.

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“…The C * -algebra or von Neumann algebra generated by Toeplitz operators with such symbols is abelian on each weighted Bergman space. For details, the reader can consult [5,6,7] and [20,21]. The inverse is true in an appropriate sense.…”

Section: Introductionmentioning

confidence: 99%

Maximal Abelian von Neumann Algebras and Toeplitz Operators with Separately Radial Symbols

Huang

2009

Integr. equ. oper. theory

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This paper mainly concerns abelian von Neumann algebras generated by Toeplitz operators on weighted Bergman spaces. Recently a family of abelian w * -closed Toeplitz algebras has been obtained (see [5,6,7,8]). We show that this algebra is maximal abelian and is singly generated by a Toeplitz operator with a "common" symbol. A characterization for Toeplitz operators with radial symbols is obtained and generalized to the high dimensional case. We give several examples for abelian von Neumann algebras in the case of high dimensional weighted Bergman spaces, which are different from the one dimensional case.

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Quasi-Radial Quasi-Homogeneous Symbols and Commutative Banach Algebras of Toeplitz Operators

Cited by 40 publications

References 7 publications

Algebras of Toeplitz Operators on the n-Dimensional Unit Ball

Algebras of Toeplitz Operators on the n-Dimensional Unit Ball

Commutative Toeplitz Banach Algebras on the Ball and Quasi-Nilpotent Group Action

Maximal Abelian von Neumann Algebras and Toeplitz Operators with Separately Radial Symbols

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