Continuous minimax theorems

TL;DR

This paper generalizes classical extremal eigenvalue theorems from matrix theory to self-adjoint elements in finite von Neumann algebras, extending min-max principles to a broader operator algebra context.

Contribution

It introduces a continuous minimax theorem for self-adjoint operators in finite von Neumann algebras, extending classical eigenvalue characterizations.

Findings

Extended extremal eigenvalue characterizations to von Neumann algebras.

Established a continuous minimax theorem for self-adjoint operators.

Connected classical matrix results with operator algebra theory.

Abstract

In classical matrix theory, there exist useful extremal characterizations of eigenvalues and their sums for Hermitian matrices (due to Ky Fan, Courant-Fischer-Weyl and Wielandt) and some consequences such as the majorization assertion in Lidskii's theorem. In this paper, we extend these results to the context of self adjoint elements of finite von Neumann algebras, and their distribution and quantile functions. This work was motivated by a lemma in a paper by Voiculescu and Bercovici, that described such an extremal characterization of the distribution of a self-adjoint operator affiliated to a finite von Neumann algebra - suggesting a possible analogue of the classical Courant-Fischer-Weyl minmax theorem, for a self adjoint operator in a finite von Neumann algebra. It is to be noted that the only von Neumann algebras considered here have separable pre-duals.