The paper studies the structure and commutative properties of von Neumann algebras induced by multiplication operators on the Bergman space of a bounded domain in the complex space C^d. We show that there is a close interplay between operator theory, geometry, and function theory.

This work is partially supported by National Natural Science Foundation of China. The authors are in debt to the referee for many valuable suggestions which make this paper more transparent and more readable. The authors would like to thank Professor Dechao Zheng at Vanderbilt University for helpful discussions while the paper was in progress.