Von Neumann Algebra On The Bergman Space, Reducing Subspaces And Related Geometric Analysis

In this thesis,we mainly consider the reducing subspaces of a multiplication operator M_φ,the von Neumann algebra W^*(φ) generated by M_φand related geometric analysis on the Bergman space L_α²(D) over the unit disk D.Since the Bergman space is defined via the area measure,the theory of L_α²(D) has tight relation with complex analysis and geometry.By von Neumann's bicommutant theorem,W^*(φ)″=W^*(φ), which shows that in some sense studying W^*(φ) is equivalent to studying its commutant algebra V^*(φ)(?)W^*(φ)′,which is also a von Neumann algebra.In the case whenφis a thin Blaschke product,by using the technique of analytic continuation and local inverses,we established the representation of unitary operators in V^*(B),which generalized Sun's result on finite Blaschke products[Sun1].And it is shown that if B is a finite Blaschke product with deg B≤6,then the number of minimal reducing subspaces of M_B is at most deg B.When deg B=2 and deg B=3,4,these results were obtained in[SW,Zhu]and[GSZZ,SZZ1]respectively.Moreover,we consider V^*(φ) in the case whenφis a holomorphic covering map from D onto a bounded domainΩin C.We give the representation of those unitary operators commuting with M_φ.Applying this result,we show that the von Neumann algebra V^*(φ) is abelian if and only if the fundamental groupÏ€₁(Ω) ofΩis abelian; if and only ifΩis conformally isomorphic to the disk,annuli or the punctured disk. Except for the above three cases,we also find that V^*(φ) is always a Typeâ…¡₁ factor, and W^*(φ) is a Typeâ…¡_∞factor.Furthermore,we generalize this result nontrivially to the case ofφbeing a holomorphic regular branched covering map,where the structure of V^*(φ) is tightly related with orbifold Riemann surface and group theory.Some deep consequences are also established.For example,there exist many infinite Blaschke products B such that M_B have as many reducing subspaces as the elements in[0,1]. This is quite different from the case of finite Blaschke product. We also study those analytic varieties in ploydisk which have the H^∞-extension property and obtain some deep results.As an application,we give a sufficient condition for the uniqueness of a three-point Pick-Nevanlinna interpolation problem in D³.