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

Abstract

Let $M$ be a type ${\rm II}$ factor and let $τ$ be the faithful positive semifinite normal trace, unique up to scalar multiples in the type ${\rm II}_\infty$ case and normalized by $τ(I)=1$ in the type ${\rm II}_1$ case. Given $A\in M^+$, we denote by $A_+=(A-I)χ_A(1,\|A\|]$ the excess part of $A$ and by $A_-=(I-A)χ_A(0,1)$ 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 $τ(A_+)\geq τ(A_-)$ 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 $τ(A_+)\geq τ(A_-)$, 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.