Inclusions and positive cones of von Neumann algebras

TL;DR

This paper investigates the inclusion relations of von Neumann algebras through associated cones in Hilbert spaces, revealing how Jordan structures and projections can be characterized in specific cases.

Contribution

It introduces a method to determine algebra inclusion via cones and recovers Jordan structures from these cones, providing new insights into von Neumann algebra structure.

Findings

Criteria for algebra inclusion based on cone properties

Recovery of Jordan structure from cones

Characterization of projections in special cases

Abstract

We consider cones in a Hilbert space associated to two von Neumann algebras and determine when one algebra is included in the other. If a cone is assocated to a von Neumann algebra, the Jordan structure is naturally recovered from it and we can characterize projections of the given von Neumann algebra with the structure in some special situations.