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Let $M$ be a type ${\rm II}$ factor and let $\tau$ be the faithful positivesemifinite normal trace, unique up to scalar multiples in the type ${\rmII}_\infty$ 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 andS. Zhang provided necessary and sufficient conditions for a positive operatorto be the sum of a finite or infinite collection of projections (notnecessarily 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 ofprojections and also sufficient if the operator is "diagonalizable". In thispaper, we prove that if $A\in M^+$ and $\tau(A_+)\geq \tau(A_-)$, then $A$ canbe written as the sum of a finite or infinite collection of projections. Thisresult answers affirmatively a question raised by V. Kaftal, P. Ng and S.Zhang.

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