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تسجيل مستخدم جديدStrong sums of projections in type ${rm II}$ factors
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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 $Ain M^+$, 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 $Ain M^+$ 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 $Ain M^+$ 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.
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