  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.
