On algebras generated by Toeplitz operators and their representations

Abstract

We study Banach and C⁎-algebras generated by Toeplitz operators acting on weighted Bergman spaces Aλ2(B2) over the complex unit ball B2⊂C2. Our key point is an orthogonal decomposition of Aλ2(B2) into a countable sum of infinite dimensional spaces, each one of which can be identified with a differently weighted Bergman space Aμ2(D) over the complex unit disk D. Moreover, all elements of the above algebras leave each of the summands in the above decomposition invariant and their restriction to each level acts as a compact perturbation of a Toeplitz operator on Aμ2(D).The symbols of the generating Toeplitz operators are chosen to be suitable extensions to B2 of families S of bounded functions on D. Symbol classes S that generate important classical commutative and non-commutative Toeplitz algebras in L(Aμ2(D)) are of particular interest. In this paper we discuss various examples. In the case of S=C(D‾) and S=C(D‾)⊗L∞(0,1) we characterize all irreducible representations of the resulting Toeplitz operator C⁎-algebras. Their Calkin algebras are described and index formulas are provided.