An Operator Theoretic Approach to Birkhoff's Problem 111

TL;DR

This paper investigates Birkhoff's problem 111 using operator theory, demonstrating that the convex hull of infinite permutation matrices cannot be achieved in any standard operator topology, and explores the structure of these matrix spaces.

Contribution

It extends Birkhoff's problem to infinite matrices using operator topologies, showing the non-existence of such a hull in these topologies and analyzing the structure of the matrix spaces.

Findings

Birkhoff's problem 111 has no solution in any operator topology.

Kendall's affirmative topology applies to all finer locally convex topologies.

The closed affine hull includes all operators with real entries.

Abstract

In 1946, Garrett Birkhoff proved that the $n\times n$ doubly stochastic matrices comprise the convex hull of the $n\times n$ permutation matrices, which in turn make up the extreme points of this polytope. He proposed his problem 111, which asks whether there exists a topology on infinite matrices for which this applies to the closed convex hull of the $\mathbb{N}\times\mathbb{N}$ permutation matrices. As Isbell showed in 1955, this equality is not achieved in the line-sum norm. In this paper, we use the domain of operator theory, and its many topologies, to improve on his negative result by showing that Birkhoff's problem is not solved in any of these topologies. In Kendall's 1960 paper on this problem, he gave an answer to the affirmative, as well as a topology for which closed convex hull comprises the doubly substochastic matrices. We also show that Kendall's secondary theorem also applies for all the locally convex Hausdorff topologies finer than than Kendall's (namely that of entry-wise convergence) which make the continuous dual of the matrix space no larger than the predual of the von Neumann algebra containing them. We then show that this is a theoretical upper limit topologies with this closure property. We also discuss the exposed points of this hull for these several topologies. Moreover, we show that, in these topologies, the closed affine hull of these permutation matrices comprise all operators with real-entry matrix coefficients.