A generalisation to Birkhoff - von Neumann theorem

TL;DR

This paper extends the Birkhoff-von Neumann theorem to a measure-theoretic setting involving measure-preserving partial isomorphisms, broadening its applicability beyond doubly stochastic matrices.

Contribution

It generalizes the classical theorem to the type II_1 setting, demonstrating that a weaker version still holds with measure-preserving partial isomorphisms.

Findings

A weaker version of the theorem is valid in the measure-theoretic context.

The classical convex hull characterization extends to measure-preserving partial isomorphisms.

The result broadens the understanding of matrix decomposition in infinite measure spaces.

Abstract

The classic Birkhoff- von Neumann theorem states that the set of doubly stochastic matrices is the convex hull of the permutation matrices. In this paper, we study a generalisation of this theorem in the type $II_1$ setting. Namely, we replace a doubly stochastic matrix with a collection of measure preserving partial isomorphisms, of the unit interval, with similar properties. We show that a weaker version of this theorem still holds.