On the Toeplitz Algebra in the Case of All Entire Functions and All Functions Holomorphic in the Unit Disc

We study the algebra generated by all Toeplitz operators on the Fréchet space of all entire functions and all functions holomorphic in the unit disk. In both cases we prove that the quotient algebra by the commutator ideal can be equipped with a locally convex topology which makes this quotient algebra algebraically and topologically isomorphic with the symbol algebra. We also show that the topology of uniform convergence on bounded sets is not a correct choice here, since the commutator ideal is dense in the algebra generated by all Toeplitz operators in this topology. However, the linear space of all Toeplitz operators with the topology of uniform convergence on bounded sets is linearly isomorphic with the symbol space. There are also some other subtle differences between the case which we study and the classical one. Our theorems provide the key step towards extending the results previously obtained for single Toeplitz operators to the elements of the algebra generated by all Toeplitz operators.