Commutative algebras generated by Toeplitz operators - Gelfand theory and spectral properties

在N.Vasilevski和W.Bauer/N.Vasilveski的一系列文章中，最近发现了由C^n中复单位球上的标准加权Bergman空间上的算子组成的新型交换Toeplitz Banach代数。这些代数只出现在更高维的环境n&gt；1中，它们还远未被完全理解。在当前的项目中，我们重点对它们进行分类和分析。更确切地说，我们的主要目的是：1.对C^N中n维复单位球B的自同构群的极大交换子群从属的交换Toeplitz Banach代数的类型进行完全分类。对文[1]中得到的代数的内部结构的一种刻画。特别地，我们打算研究它们的Gelfand理论，确定极大理想空间和根。这些代数中有哪些是半单的？3.作为2中结果的一个应用，我们计划决定这些代数的“谱不变性”问题。此外，我们相信我们可以对谱理论及其元素的Fredholm性有更深的理解。对于更一般的有界对称域上的Toeplitz算子或单位球上不同的泛函Hilbert空间(例如调和的L^2-函数)，也可以提出类似的问题。在这些情况下，1.-3.的完整解决方案似乎相当具有挑战性。

In a series of papers by N. Vasilevski and W. Bauer/N. Vasilveski new types of commutative Toeplitz Banach algebras consisting of operators on the standard weighted Bergman spaces over the complex unit ball in C^n have been discovered recently. These algebras only show up in the higher dimensional setting n>1 and they are still far from being completely understood. In the current project we focus on their classification and analysis. More precisely, the following tasks are our main objectives:1. The full classification of the types of commutative Toeplitz Banach algebras that are sub-ordinate to the maximal commutative subgroups of the automorphism group of the n-dimensional complex unit ball B in C^n.2. A description of the internal structure of the algebras obtained in 1. In particular, we intend to study their Gelfand theory, determine the maximal ideal spaces and the radical. Which of these algebras are semi-simple?3. As an application of the results in 2. we plan to decide the question of the "spectral invariance" of these algebras. Moreover we believe that we can reach a deeper understanding of the spectral theory and the Fredholm property of their elements. Similar questions can be asked in the case of Toeplitz operators on more general bounded symmetric domains or for different functional Hilbert spaces (e.g. harmonic L^2-functions) over the unit ball. In these cases a complete solution of 1. - 3. seems quite challenging.