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

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.