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 ϕ, we will show that the largest C⁎-algebra in the commutant of the multiplication operator Mϕ by ϕ on the Bergman space is finite dimensional, and its dimension equals the number of connected components of the Riemann surface of ϕ−1∘ϕ over the unit disk. If the order of the Blaschke product ϕ is less than or equal to eight, then every C⁎-algebra contained in the commutant of Mϕ is abelian and hence the number of minimal reducing subspaces of Mϕ equals the number of connected components of the Riemann surface of ϕ−1∘ϕ over the unit disk.