Multiplication operators defined by covering maps on the Bergman space: The connection between operator theory and von Neumann algebras

Abstract

In this paper, we combine methods of complex analysis, operator theory and conformal geometry to construct a class of Type II factors in the theory of von Neumann algebras, which arise essentially from holomorphic coverings of bounded planar domains. One will see how types of such von Neumann algebras are related to algebraic topology of planar domains. As a result, the paper establishes a fascinating connections to one of the long-standing problems in free group factors. An interplay of analytical, geometrical, operator and group theoretical techniques is intrinsic to the paper.

Reducing Subspaces of Some Multiplication Operators on the Bergman Space over Polydisk

“…Finally, an affirmative answer to the conjecture is given by Douglas et al [11]. Furthermore, when is an infinite Blaschke product, some relative results are obtained by Guo and Huang in [12,13].…”

Geometric constructions of thin Blaschke products and reducing subspace problem

“…However, there do exist interpolating Blaschke products which are also covering maps, see [Cow1,Theorem 6] or [GH2,Example 3.6]. Below, we denote by ∆(z, r) the pseudohyperbolic disk centered at z with radius r; that is,…”

“…Proposition 6.9 is sharp in the sense that it fails for interpolating Blaschke products, especially for those holomorphic covering maps B : D → D − F , with F a discrete subset of D. For details, refer to [GH2,Examples 3.5 and 3.6].…”

An Introduction to Hilbert Module Approach to Multivariable Operator Theory

“…A few of these are: (1) The classification program for reducing subspaces of multiplication by Blaschke products on the Bergman space, by Zhu, Guo, Douglas, Sun, Wang, Putinar. (see [DoSuZ11], [DoPuW12], [GuH11]). (2) Extensions of Hilbert modules by Carlson, Clark, Foias, Guo, Didas and Eschmeier (see [DiEs06], [CaCl95], [CaCl97], [Gu99]).…”