Analytic continuation of Toeplitz operators and commuting families of $C^{\ast }-$algebras

We consider the Toeplitz operators on the weighted Bergman spaces over the unit ball ${\mathbb{B}}^n$ and their analytic continuation. We proved the commutativity of the $C^{\ast }-$algebras generated by the analytic continuation of Toeplitz operators with a special class of symbols that satisfy an invariant property, and we showed that these commutative $C^{\ast }-$algebras with symbols invariant under compact subgroups of $SU(n,1)$ are completely characterized in terms of restriction to multiplicity free representations. Moreover, we extended the restriction principal to the analytic continuation case for suitable maximal abelian subgroups of $SU(n,1)$, we obtained the generalized Segal-Bargmann transform and we showed that it acts as a convolution operator. Furthermore, we proved that Toeplitz operators are unitarly equivalent to a convolution operator and we provided integral formulas for their spectra.