TY - JOUR

T1 - Commutative Toeplitz Banach Algebras on the Ball and Quasi-Nilpotent Group Action

AU - Bauer, Wolfram

AU - Vasilevski, Nikolai

N1 - Funding Information: W. Bauer has been supported by an “Emmy-Noether scholarship” of Forschungsgemeinschaft). N. Vasilevski has been partially supported Project 102800, México. Copyright: Copyright 2012 Elsevier B.V., All rights reserved.

PY - 2011/12/1

Y1 - 2011/12/1

N2 - 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 quasi-hyperbolic 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.

AB - 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 quasi-hyperbolic 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.

KW - Commutative Banach algebra

KW - Quasi-homogeneous

KW - Quasi-nilpotent

KW - Toeplitz operator

KW - Weighted Bergman space

UR - http://www.scopus.com/inward/record.url?scp=84855651342&partnerID=8YFLogxK

U2 - 10.1007/s00020-011-1927-7

DO - 10.1007/s00020-011-1927-7

M3 - Article

AN - SCOPUS:84855651342

VL - 72

SP - 223

EP - 240

JO - Integral Equations and Operator Theory

JF - Integral Equations and Operator Theory

SN - 0378-620X

IS - 2

ER -