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This paper presents necessary and sufficient conditions for a positivebounded operator on a separable Hilbert space to be the sum of a finite orinfinite collection of projections (not necessarily mutually orthogonal), withthe sum converging in the strong operator topology if the collection isinfinite. A similar necessary condition is given when the operator and theprojections are taken in a type II von Neumann factor, and the condition isproven to be also sufficient if the operator is "diagonalizable". A simplernecessary and sufficient condition is given in the type III factor case.

Strong sums of projections in von Neumann factors