Weakly Ordered A-Commutative Partial Groups of Linear Operators Densely Defined on Hilbert Space

Jirí Janda

doi:10.14311/1807

Abstract

The notion of a generalized effect algebra is presented as a generalization of effect algebra for an algebraic description of the structure of the set of all positive linear operators densely defined on a Hilbert space with the usual sum of operators. The structure of the set of not only positive linear operators can be described with the notion of a weakly ordered partial commutative group (wop-group).Due to the non-constructive algebraic nature of the wop-group we introduce its stronger version called a weakly ordered partial a-commutative group (woa-group). We show that it also describes the structure of not only positive linear operators.

Highlights

The notion of an effect algebra was presented by Foulis and Bennett in [3]
The definition was motivated by giving an algebraic description of positive self-adjoint linear operators between the zero and the identity operator in a complex Hilbert space H
[6] Paseka and Janda introduced the structure of a weakly ordered partial commutative group. They showed that the set of all linear operators on complex Hilbert space H with the usual sum, which is restricted to the same domain for unbounded operators, possesses this structure

Summary

Introduction

The notion of an effect algebra was presented by Foulis and Bennett in [3]. The definition was motivated by giving an algebraic description of positive self-adjoint linear operators between the zero and the identity operator in a complex Hilbert space H. In [6] Paseka and Janda introduced the structure of a weakly ordered partial commutative group (shortly a wop-group) They showed that the set of all linear operators on complex Hilbert space H with the usual sum, which is restricted to the same domain for unbounded operators (partial operation ⊕D), possesses this structure. Wop-groups have only a non-constructive associativity (the equation holds if and only if both sides are defined) It has been shown [4] that the set of all linear operators has generally stronger algebraic properties. This was a motivation for introducing the notion of a weakly ordered partial a-commutative group (woa-group) where the associative law is more constructive. We present a construction showing that every generalized effect algebra is a positive cone of some woa-group

Preliminaries

Hilbert spaces