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

Abstract

Let $M$ be a type $\mathrm{II}$ factor and let $\tau$ 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 $A\in M^{+}$, 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 $A$. 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 $\mathrm{I}$ and type $\mathrm{III}$ factors. For type $\mathrm{II}$ factors, V. Kaftal, P. Ng and S. Zhang proved that $\tau (A_{+})\ge \tau (A_{-})$ is a necessary condition for an operator $A\in M^{+}$ 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 $\tau (A_{+})\ge \tau (A_{-})$, 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.

10 pages