Bicommutants of reduced unbounded operator algebras

TL;DR

This paper investigates the properties of unbounded bicommutants of reduced O*-algebras, aiming to identify conditions that preserve algebraic structure under reduction and exploring implications for conditional expectations.

Contribution

It provides new conditions under which the reduction of a GW*-algebra remains a GW*-algebra, advancing understanding of unbounded operator algebra structure.

Findings

Conditions for the reduction of a GW*-algebra to be a GW*-algebra.

Application to the existence of conditional expectations on O*-algebras.

Analysis of the unbounded bicommutant structure in reduced algebras.

Abstract

The unbounded bicommutant $(\mathfrak M_{E'})''$ of the {\em reduction} of an O*-algebra $\MM$ via a given projection $E'$ weakly commuting with $\mathfrak M$ is studied, with the aim of finding conditions under which the reduction of a GW*-algebra is a GW*-algebra itself. The obtained results are applied to the problem of the existence of conditional expectations on O*-algebras.