The topologies of local convergence in measure on the algebra of measurable operators

Abstract

Given a von Neumann algebra $M$ of operators on a Hilbert space $H$ and a faithful normal semifinite trace $\tau$ on $M$, denote by $S(M, \tau)$ the *-algebra of $\tau$-measurable operators. We obtain a sufficient condition for the positivity of an hermitian operator in $S(M, \tau)$ in terms of the topology $t_{\tau l}$ of $\tau$-local convergence in measure. We prove that the *-ideal $F(M, \tau)$ of elementary operators is $t_{\tau l}$-dense in $S(M, \tau)$. If $t_{\tau}$ is locally convex then so is $t_{\tau l}$; if $t_{\tau l}$ is locally convex then so is the topology $t_{w \tau l}$ of weakly $\tau$-local convergence in measure. We propose some method for constructing F-normed ideal spaces, henceforth F-NIPs, on $(M, \tau)$ starting from a prescribed F-NIP and preserving completeness, local convexity, local boundedness, or normability whenever present in the original. Given two F-NIPs $X$ and $Y$ on $(M, \tau)$, suppose that $AX\subseteq Y$ for some operator $A \in S(M, \tau)$. Then the multiplier $M_AX = AX$ acting as $M_A : X \to Y$ is continuous. In particular, for $X\subseteq Y$ the natural embedding of $X$ into $Y$ is continuous. We inspect the properties of decreasing sequences of F-NIPs on $(M, \tau)$.