Abstract. By the classical Brown, Halmos (1964) result, there is no commutative $C^∗$-algebra generated by Toeplitz operators, with non-trivial symbols, acting on the Hardy space $H^2(S^1)$, while there are only two non-trivial commutative Banach algebras generated by Toeplitz operators. For one of them symbols are analytic, and are conjugate analytic, for the other.
At the same time, as it was observed recently, there are many non-trivial commutative $C^∗$-algebras generated by Toeplitz operators, acting on the Bergman space over the unit disk. Moreover, for a multidimensional case of the weighted Bergman space $A^2_{\lambda}(B^n)$, apart of a wide variety of commutative $C^∗$-algebras, there exist many commutative Banach algebras, all of them are generated by Toeplitz operators with symbols from different specific classes.
The aim of the talk is to clarify the situation for a multidimensional Hardy space $H^2(S^{2n−1})$ case.
We present an universal approach that permits us to unhide and describe both commutative $C^∗$ and Banach algebras generated by Toeplitz operators on $H^2(S^{2n−1})$, as well as to describe some non-commutative $C^∗$-algebras. In the latter case we characterize, among others, their irreducible representa- tions and spectral properties of the corresponding Toeplitz operators.