Let $\mathcal{M}$ be a type II factor and let τ be the faithful positive semifinite normal trace, unique up to scalar multiples in the type ${\mathrm{II}}_{\infty }$ case and normalized by $\tau (I)=1$ in the type ${\mathrm{II}}_1$ case. Given $A\in {\mathcal{M}}^{+}$, we denote by $A_{+}=(A-I){\chi }_A(1,\Vert A\Vert ]$ the excess part of A and by $A_{-}=(I-A){\chi }_A(0,1)$ the defect part of A. In [6], 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 I and type III factors. For type II factors, V. Kaftal, P. Ng and S. Zhang proved that $\tau (A_{+})\ge \tau (A_{-})$ is a necessary condition for an operator $A\in {\mathcal{M}}^{+}$ 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 $A\in {\mathcal{M}}^{+}$ and $\tau (A_{+})\ge \tau (A_{-})$, then A can be written as the sum of a finite or infinite collection of projections. This result answers affirmatively Question 5.4 of [6].

让 $\mathcal{中号}$是II型因子，令τ是忠实的正半正法线，在该类型中的标量倍数之前是唯一的${\mathrm{II}}_{\infty }$ 情况并通过 $\tau （\mathrm{一世}）=\mathrm{1个}$ 在类型 ${\mathrm{II}}_{\mathrm{1个}}$案件。给定$\mathrm{一种}\in {\mathcal{中号}}^{+}$，我们用 ${\mathrm{一种}}_{+}=（\mathrm{一种}-\mathrm{一世}）{\chi }_{\mathrm{一种}}（\mathrm{1个}，\Vert \mathrm{一种}\Vert ]$的超出部分，阿和由${\mathrm{一种}}_{-}=（\mathrm{一世}-\mathrm{一种}）{\chi }_{\mathrm{一种}}（0，\mathrm{1个}）$A的缺陷部分。在[6]中，V。Kaftal，P。Ng和S. Zhang为正算子提供了I型和III型因子的有限或无限集合（不一定相互正交）的总和的必要条件和充分条件。 。对于II型因子，V。Kaftal，P。Ng和S. Zhang证明了$\tau （{\mathrm{一种}}_{+}）\ge \tau （{\mathrm{一种}}_{-}）$ 是操作员的必要条件 $\mathrm{一种}\in {\mathcal{中号}}^{+}$可以写成有限的或无限的投影集合的总和，如果运算符是“对角线化的”，则也足够。在本文中，我们证明$\mathrm{一种}\in {\mathcal{中号}}^{+}$ 和 $\tau （{\mathrm{一种}}_{+}）\ge \tau （{\mathrm{一种}}_{-}）$，则A可以写为投影的有限或无限集合之和。该结果肯定地回答了[6]的问题5.4。