Finite sums of projections in von Neumann algebras (Q2838070)

In this paper, the authors consider two problems in the context of general von Neumann algebras. That is, which positive operators are finite linear combinations of projections with positive coefficients (called positive combinations of projections) and which positive operators are finite sums of projections. Some necessary and sufficient conditions for a positive element \(a\) in a von Neumann algebra \(M\) with properly infinite range projection \(R_a\) to be a positive combination of projections in \(M\) are given. In particular, the authors prove that, in a \(\sigma\)-finite von~Neumann algebra \(M\), a positive element \(a\) with properly infinite range projection \(R_a\) is a positive combination of projections if and only if the essential norm \(\|a\|_e\) with respect to the closed two-sided ideal \(J(M)\) generated by the finite projections on \(M\) does not vanish. If \(\|a\|_e>1\), then \(a\) is a finite sum of projections. Both results are extended to general properly infinite von~Neumann algebras in terms of central essential spectra. Meanwhile, the authors also provide a necessary condition or a positive operator \(a\) to be a finite sum of projections in terms of the principal ideals generated by the excess part \(a_+:=(a-I)\chi_a(1,\infty)\) and the defect part \(a_-:=(I-a)\chi_a(0,1)\) of \(a\). Finally, they prove that, in a type \(II_1\) factor, a sufficient condition for a positive diagonalizable operator to be a finite sum of projections is that \(\tau(a_+)>\tau(a_-)\).