Algebras of Toeplitz operators on the $n$-dimensional unit ball

We study $C^{\ast }$-algebras generated by Toeplitz operators acting on the standard weighted Bergman space ${\mathcal{A}}_{\lambda }^2({\mathbb{B}}^n)$ over the unit ball ${\mathbb{B}}^n$ in ${\mathbb{C}}^n$. The symbols $f_{ac}$ of generating operators are assumed to be of a certain product type. By choosing $a$ and $c$ in different function algebras ${\mathcal{S}}_a$ and ${\mathcal{S}}_c$ over lower dimensional unit balls ${\mathbb{B}}^{\mathrm{\ell }}$ and ${\mathbb{B}}^{n-\mathrm{\ell }}$, respectively, and by assuming the invariance of $a\in $\mathcal{S}$a$undersometorusactionweobtain$C^*$-algebras$\boldsymbol{$\mathcal{T}$}{\lambda}($\mathcal{S}$_a, $\mathcal{S}$_c)$ whose structural properties can be described. In the case of $k$-quasi-radial functions ${\mathcal{S}}_a$ and bounded uniformly continuous or vanishing oscillation symbols ${\mathcal{S}}_c$ we describe the structure of elements from the algebra ${\mathcal{T}}_{\lambda }({\mathcal{S}}_a,{\mathcal{S}}_c)$, derive a list of irreducible representations of ${\mathcal{T}}_{\lambda }({\mathcal{S}}_a,{\mathcal{S}}_c)$, and prove completeness of this list in some cases. Some of these representations originate from a `quantization effect’, induced by the representation of ${\mathcal{A}}_{\lambda }^2({\mathbb{B}}^n)$ 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.