We study C∗-algebras generated by Toeplitz operators acting on the standard weighted Bergman space Aλ2(Bn) over the unit ball Bn in Cn. The symbols fac of generating operators are assumed to be of a certain product type, see (\ref{Introduction_form_of_the_symbol}). By choosing a and c in different function algebras Sa and Sc over lower dimensional unit balls Bℓ and Bn−ℓ, respectively, and by assuming the invariance of a∈Sa under some torus action we obtain C∗-algebras Tλ(Sa,Sc) whose structural properties can be described. In the case of k-quasi-radial functions Sa and bounded uniformly continuous
or vanishing oscillation symbols Sc we describe the structure of elements from the algebra Tλ(Sa,Sc), derive a list of irreducible
representations of Tλ(Sa,Sc), and prove completeness of this list in some cases. Some of these representations originate from a ``quantization effect'', induced
by the representation of Aλ2(Bn) as the direct sum of Bergman spaces over a lower dimensional unit ball with growing weight parameter. As an application we derive the
essential spectrum and index formulas for matrix-valued operators