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

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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.