Speaker: Nikolai L. Vasilevski, CINVESTAV, Mexico City

Abstract: The talk is intended for a wide audience, not necessarily consisting of experts in the theory of Toeplitz operators, and is a review of the results on the description of algebras generated by Toeplitz operators. We begin with a somewhat surprising and unpredictable result on the existence of a large class of non-isomorphic commutative C∗-algebras generated by Toeplitz operators. As it turned out, their symbols must be invariant under the action of maximal Abelian subgroups of the biholomorphisms of the unit ball.

The next surprise was the discovery of a large number of Banach (not C∗) algebras, which turned out to be, as a rule, not semisimple. The problem here is to find a compact set of maximal ideals and to describe the radical.

Finally we consider non-commutative C∗-algebras generated by Toeplitz operators whose symbols are invariant under the action of a subgroup of some maximal Abelian group of biholomorphisms. It turned out that different types of action of the same subgroup lead to completely different properties of the corresponding algebras.

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