Topological algebras of measurable and locally measurable operators

M. A. Muratov ; V. I. Chilin

Contemporary Mathematics. Fundamental Directions, Proceedings of the Crimean autumn mathematical school-symposium, Tome 61 (2016), pp. 115-163

Résumé

In this paper, we review the results on topological $*$-algebras $S(\mathcal M)$, $S(\mathcal M,\tau)$ and $LS(\mathcal M)$ of measurable, $\tau$-measurable, and locally measurable operators affiliated with the von Neumann algebra $\mathcal M$. Also we consider relations between these algebras for different classes of von Neumann algebras and establish the continuity of operator-valued functions with respect to local convergence in measure. We describe maximal commutative $*$-subalgebras of the algebra $LS(\mathcal M)$ as well.