Integer operators in finite von Neumann algebras

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.