Algebras of Toeplitz Operators on the n-Dimensional Unit Ball

We study \(C^*\)-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, see (1.1). By choosing a and c in different function algebras \(\mathcal {S}_a\) and \(\mathcal {S}_c\) over lower dimensional unit balls \({\mathbb {B}}^{\ell }\) and \({\mathbb {B}}^{n-\ell }\), respectively, and by assuming the invariance of \(a\in \mathcal {S}_a\) under some torus action we obtain \(C^*\)-algebras \(\varvec{\mathcal {T}}_{\lambda }(\mathcal {S}_a, \mathcal {S}_c)\) of 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 \(\varvec{\mathcal {T}}_{\lambda }(\mathcal {S}_a, \mathcal {S}_c)\), derive a list of irreducible representations of \(\varvec{\mathcal {T}}_{\lambda }(\mathcal {S}_a, \mathcal {S}_c)\), 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.