Anatomy of the C ∗ - algebra generated by Toeplitz operators with piece - wise continuous symbols ∗

We study the structure of the C∗-algebra generated by Toeplitz operators with piece-wise continuous symbols, putting a special emphasis to Toeplitz operators with unbounded symbols. We show that none of a finite sum of finite products of the initial generators is a compact perturbation of a Toeplitz operator. At the same time the uniform closure of the set of such sum of products contains a huge amount of Toeplitz operators with bounded and unbounded symbols drastically different from symbols of the initial generators. 1 Preliminaries In the paper we continue the detailed study of the C∗-algebra generated by Toeplitz operators Ta with piece-wise continuous symbols a acting on the Bergman space A2(D) on the unit disk D in C, which was initiated in [4, 6]. We start by recalling of the necessary definitions and results of [4]. Let D be the unit disk on the complex plane and γ = ∂D be its boundary. Consider the space L2(D) with the standard Lebesgue plane measure dv(z) = dxdy, z = x+ iy ∈ D, and ∗This work was partially supported by CONACYT Project 60160, México.