Convergence in measure and $\tau$-compactness of $\tau$-measurable operators, affiliated with a semifinite von Neumann algebra

Let $ \tau $ be a faithful normal semifinite trace on a von Neumann algebra. We establish the Leibniz criterion for sign-alternating series of $ \tau $-measurable operators. An analogue of the criterion of “sandwich” convergence of series for $ \tau $-measurable operators is obtained. We prove a refinement of this criterion for the $ \tau $-compact case. In terms of measure convergence topology, the criterion of $ \tau $-compactness of an arbitrary $ \tau $-measurable operator is established. We also give a sufficient condition of 1) $ \tau $-compactness of the commutator of a $ \tau $-measurable operator and a projection; 2) convergence of $ \tau$-measurable operator and projection commutator sequences to the zero operator in the measure $ \tau $.