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

在N. Vasilevski和W. Bauer/N。最近在C^n的复单位球上的标准加权Bergman空间上发现了由算子组成的新类型的Vasilveski交换Toeplitz Banach代数。这些代数只出现在高维环境中它们还远没有被完全理解。在当前的项目中，我们主要对它们进行分类和分析。更确切地说，我们的主要目标是：C^n.2中n维复单位球B的自同构群的最大可交换子群所从属的可交换Toeplitz Banach代数类型的完全分类对第1节中得到的代数的内部结构的描述。特别地，我们打算研究他们的Gelfand理论，确定极大理想空间和根。这些代数中哪一个是半简单的？作为2中结果的应用。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光谱理论及其元素的弗雷德霍姆性质有更深入的了解。类似的问题可以在更一般的有界对称域上的Toeplitz算子或在单位球上的不同的泛函Hilbert空间（例如调和L^2函数）上提出。在这种情况下，完全解是1。- 3。看起来很有挑战性。

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.