LAUSR

Abstract We study Banach and C ⁎ -algebras generated by Toeplitz operators acting on weighted Bergman spaces A λ 2 ( B 2 ) over the complex unit ball B 2 ⊂ C 2 . Our key point is an orthogonal decomposition of A λ 2 ( B 2 ) into a countable sum of infinite dimensional spaces, each one of which can be identified with a differently weighted Bergman space A μ 2 ( 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 A μ 2 ( D ) . The symbols of the generating Toeplitz operators are chosen to be suitable extensions to B 2 of families S of bounded functions on D . Symbol classes S that generate important classical commutative and non-commutative Toeplitz algebras in L ( A μ 2 ( 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.