A tuple of multiplication operators defined by twisted holomorphic proper maps

TL;DR

This paper investigates von Neumann algebras generated by tuples of multiplication operators from holomorphic proper maps on higher-dimensional domains, revealing their structure, abelian properties, and connections to Riemann manifolds.

Contribution

It introduces a detailed analysis of von Neumann algebras associated with these operators, highlighting their geometric and algebraic properties in higher dimensions.

Findings

Von Neumann algebras can be abelian or non-abelian depending on the maps.

The structure of these algebras is closely related to the geometry of Riemann manifolds.

Examples illustrate the diverse behaviors of the algebras in different settings.

Abstract

This paper mainly concerns the von Neumann algebras induced by a tuple of multiplication operators on Bergman spaces which arise essentially from holomorphic proper maps over higher dimensional domains. We study the structures and abelian properties of the related von Neumann algebras, and in interesting cases they turns out to be tightly related to a Riemann manifold. There is a close interplay between operator theory, geometry and complex analysis. Many examples are presented.