On a canonical lattice structure on the effect algebra of a von Neumann algebra

TL;DR

This paper introduces a new spectral order on the selfadjoint part of a von Neumann algebra, turning it into a boundedly complete lattice and analyzing the conditions under which the effect algebra maps onto the projection lattice.

Contribution

It defines a novel spectral order on R_sa that extends the usual order on projections and characterizes when the effect algebra maps onto the projection lattice via the range projection.

Findings

R_sa becomes a boundedly complete lattice under the spectral order

The effect algebra E(R) is a complete lattice

The mapping A --> R(A) is a lattice homomorphism iff R is finite

Abstract

Let R be a von Neumann algebra acting on a Hilbert space H and let R_sa be the set of selfadjoint elements of R. It is well known that R_sa is a lattice with respect to the usual partial order ≤ if and only if R is abelian. We define and study a new partial order on R_sa, the spectral order ≤_s, which extends ≤ on projections, is coarser than the usual one, but agrees with it on abelian subalgebras, and turns R_sa into a boundedly complete lattice. The effect algebra E(R) := {A | 0 ≤ A ≤ I} is then a complete lattice and we show that the mapping A --> R(A), where R(A) denotes the range projection of A, is a homomorphism from the lattice E(R) onto the projection lattice P(R) of A if and only if R is a finite von Neumann algebra.