Convergence of integrable operators affiliated to a finite von Neumann algebra

In the Banach space $L_1(\mathcal M,\tau)$ of operators integrable with respect to a tracial state $\tau$ on a von Neumann algebra $\mathcal M$, convergence is analyzed. A notion of dispersion of operators in $L_2(\mathcal M,\tau)$ is introduced, and its main properties are established. A convergence criterion in $L_2(\mathcal M,\tau)$ in terms of the dispersion is proposed. It is shown that the following conditions for $X\in L_1(\mathcal M,\tau)$ are equivalent: (i) $\tau (X)=0$, and (ii) $\|I+zX\|_1\geq 1$ for all $z\in\mathbb C$. A. R. Padmanabhan's result (1979) on a property of the norm of the space $L_1(\mathcal M,\tau)$ is complemented. The convergence in $L_2(\mathcal M,\tau)$ of the imaginary components of some bounded sequences of operators from $\mathcal M$ is established. Corollaries on the convergence of dispersions are obtained.