Strong sums of projections in type ${\rm II}$ factors

TL;DR

This paper proves that in type II factors, any positive operator satisfying a specific trace inequality can be expressed as a sum of projections, extending previous results and answering an open question.

Contribution

It establishes that the trace inequality condition is sufficient for representing positive operators as sums of projections in type II factors, generalizing earlier partial results.

Findings

Operators with $ au(A_+) \u2265 au(A_-)$ can be decomposed into sums of projections.

The result applies to both finite and infinite sums.

Answers an open question by confirming sufficiency of the trace condition.

Abstract

Let $M$ be a type ${\rm II}$ factor and let $\tau$ be the faithful positive semifinite normal trace, unique up to scalar multiples in the type ${\rm II}_\infty$ case and normalized by $\tau(I)=1$ in the type ${\rm II}_1$ case. Given $A\in M^+$, we denote by $A_+=(A-I)\chi_A(1,\|A\|]$ the excess part of $A$ and by $A_-=(I-A)\chi_A(0,1)$ the defect part of $A$. V. Kaftal, P. Ng and S. Zhang provided necessary and sufficient conditions for a positive operator to be the sum of a finite or infinite collection of projections (not necessarily mutually orthogonal) in type ${\rm I}$ and type ${\rm III}$ factors. For type ${\rm II}$ factors, V. Kaftal, P. Ng and S. Zhang proved that $\tau(A_+)\geq \tau(A_-)$ is a necessary condition for an operator $A\in M^+$ which can be written as the sum of a finite or infinite collection of projections and also sufficient if the operator is "diagonalizable". In this paper, we prove that if $A\in M^+$ and $\tau(A_+)\geq \tau(A_-)$, then $A$ can be written as the sum of a finite or infinite collection of projections. This result answers affirmatively a question raised by V. Kaftal, P. Ng and S. Zhang.