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) o 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) o phi over the unit disk. (c) 2010 Elsevier Inc. All rights reserved.