Commutative Toeplitz Banach Algebras on the Ball and Quasi-Nilpotent Group Action
Commutative Toeplitz Banach Algebras on the Ball and Quasi-Nilpotent Group Action
复制标题
DOI：
10.1007/s00020-011-1927-7
复制
发表时间：
2012-02
影响因子：
0.8
通讯作者：
W. Bauer;N. Vasilevski
中科院分区：
文献类型：
--
作者：
W. Bauer;N. Vasilevski
研究单位球上Toeplitz算子生成的交换C*代数，证明给定单位球双全同态的最大交换子群，在该子群作用下符号不变的Toeplitz算子生成的C*代数在每个标准加权Bergman空间上是可交换的。这些子群有五种不同的成对非共轭模型类：拟椭圆、拟抛物线、拟双曲、幂零和拟幂零。最近，Vasilevski (Integr Equ Oper Theory. 66:141–152, 2010) 观察到，还有许多其他非几何定义的符号类别，它们在每个加权伯格曼空间上生成交换托普利茨算子代数。这些符号类隶属于拟椭圆群，相应的交换算子代数是Banach，并且扩展到C*-代数它们变得不可交换。然后将这些结果扩展到符号类别，从属于拟双曲群和拟抛物线群。本文证明了符号隶属于拟幂零群的托普利茨算子的类似交换性结果。同时我们推测，除了已知的C*代数情况外，不再有由托普利茨算子生成的新的巴拿赫代数，其符号从属于零幂群，并且在每个加权伯格曼空间上是可交换的。
Studying commutativeC*-algebras generated by Toeplitz operators on the unit ball it was proved that, given a maximal commutative subgroup of biholomorphisms of the unit ball, theC*-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, nilpotentandquasi-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 thequasi-ellipticgroup, the corresponding commutative operator algebras wereBanach, and being extended toC*-algebras they became non-commutative. These results were extended then to the classes of symbols, subordinated to thequasi-hyperbolicandquasi-parabolicgroups. In this paper we prove the analogous commutativity result for Toeplitz operators whose symbols are subordinated to thequasi-nilpotentgroup. At the same time we conjecture that apart from the knownC*-algebra cases there are no more new Banach algebras generated by Toeplitz operators whose symbols are subordinated to thenilpotentgroup and which are commutative on each weighted Bergman space.