Intersections of Schubert varieties and eigenvalue inequalities in an arbitrary finite factor

TL;DR

This paper extends classical eigenvalue inequalities known in hyperfinite factors to arbitrary finite factors, providing a broader understanding of spectral properties in operator algebras.

Contribution

It proves that eigenvalue inequalities analogous to Horn inequalities hold in any finite factor, not just hyperfinite ones, and relates this to Connes' embedding problem.

Findings

Eigenvalue inequalities hold in arbitrary finite factors.

The result generalizes known inequalities from hyperfinite to all finite factors.

Connections to Connes' embedding problem are established.

Abstract

It is known that the eigenvalues of selfadjoint elements a,b,c with a+b+c=0 in the factor R^omega (ultrapower of the hyperfinite II1 factor) are characterized by a system of inequalities analogous to the classical Horn inequalities of linear algebra. We prove that these inequalities are in fact true for elements of an arbitrary finite factor. A matricial (`complete') form of this result is equivalent to an embedding question formulated by Connes.