Strong sums of projections in type II factors

Let M be a type II factor and let τ be the faithful positive semifinite normal trace, unique up to scalar multiples in the type II ∞ case and normalized by τ ( I ) = 1 in the type II 1 case. Given A ∈ M + , we denote by A + = ( A − I ) χ A ( 1 , ‖ A ‖ ] the excess part of A and by A − = ( I − A ) χ 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 τ ( A + ) ≥ τ ( A − ) is a necessary condition for an operator A ∈ 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 ∈ M + and τ ( A + ) ≥ τ ( 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] .