Conditions of coincidence of central extensions of von Neumann algebras and algebras of measurable operators

TL;DR

This paper investigates the conditions under which the central extension of a von Neumann algebra coincides with its algebra of measurable operators, providing a characterization for specific classes of these algebras.

Contribution

It characterizes when the central extension of a von Neumann algebra equals the algebra of measurable operators, clarifying the structure of such algebras.

Findings

Identifies classes of von Neumann algebras where $E(M) = S(M)$

Provides conditions for $E(M)$ to coincide with $S(M, au)$

Enhances understanding of the structure of measurable operator algebras

Abstract

Given a von Neumann algebra $M$ we consider the central extension $E(M)$ of $M.$ We describe class of von Neumann algebras $M$ for which the algebra $E(M)$ coincides with the algebra $S(M)$ -- the algebra of all measurable operators with respect to $M,$ and with $S(M, \tau)$ -- the algebra of all $\tau$-measurable operators with respect to $M.$