Strong sums of projections in von Neumann factors

TL;DR

This paper characterizes when positive operators in von Neumann factors can be expressed as sums of projections, providing necessary and sufficient conditions across different factor types, with special cases for diagonalizable operators.

Contribution

It establishes new necessary and sufficient conditions for representing positive operators as sums of projections in various von Neumann factors, including type II and III.

Findings

Conditions for sums of projections in type II factors

Conditions for sums of projections in type III factors

Characterization of diagonalizable operators in this context

Abstract

This paper presents necessary and sufficient conditions for a positive bounded operator on a separable Hilbert space to be the sum of a finite or infinite collection of projections (not necessarily mutually orthogonal), with the sum converging in the strong operator topology if the collection is infinite. A similar necessary condition is given when the operator and the projections are taken in a type II von Neumann factor, and the condition is proven to be also sufficient if the operator is "diagonalizable". A simpler necessary and sufficient condition is given in the type III factor case.