Multiplication operators on the Bergman space via analytic continuation

In this paper, using the group-like property of local inverses of a finite Blaschke product $\phi$, we will show that the largest $C^*$-algebra in the commutant of the multiplication operator $M_{\phi}$ by $\phi$ on the Bergman space is finite dimensional, and its dimension equals the number of connected components of the Riemann surface of $\phi^{-1}\circ\phi$ over the unit disk. If the order of the Blaschke product $\phi$ is less than or equal to eight, then every $C^*$-algebra contained in the commutant of $M_{\phi}$ is abelian and hence the number of minimal reducing subspaces of $M_{\phi}$ equals the number of connected components of the Riemann surface of $\phi^{-1}\circ\phi$ over the unit disk.