Integer operators in finite von Neumann algebras

TL;DR

This paper introduces and studies integer operators in finite von Neumann algebras, linking their spectral properties to classical potential theory and equidistribution of algebraic integers, with implications for spectral analysis of certain operators.

Contribution

It defines integer operators in finite von Neumann algebras and connects their spectral characteristics to potential theory and algebraic integer distribution, extending previous mathematical frameworks.

Findings

Characterization of integer operators with spectrum of logarithmic capacity ≤ 1

Connection between integer operators and classical potential theory

Relation to recent algebraic integer equidistribution results

Abstract

Motivated by the study of spectral properties of self-adjoint operators in the integral group ring of a sofic group, we define and study integer operators. We establish a relation with classical potential theory and in particular the circle of results obtained by M. Fekete and G. Szeg"o. More concretely, we use results by R. Rumely on equidistribution of algebraic integers to obtain a description of those integer operator which have spectrum of logarithmic capacity less or equal to one. Finally, we relate the study of integer operators to a recent construction by B. and L. Petracovici and A. Zaharescu.