Three characterisations of the sequential product

TL;DR

This paper characterizes the standard sequential product in operator algebras using three additional properties, providing a deeper understanding of its structure on von Neumann and Euclidean Jordan algebras.

Contribution

It introduces three new properties that uniquely characterize the standard sequential product on certain algebraic structures.

Findings

Standard sequential product characterized by invariance, symmetry, and invertibility preservation.

Convex σ-sequential effect algebras correspond to spectral order unit spaces.

Provides a new framework for understanding sequential effects in operator algebras.

Abstract

It has already been established that the properties required of an abstract sequential product as introduced by Gudder and Greechie are not enough to characterise the standard sequential product $a\circ b = \sqrt{a}b\sqrt{a}$ on an operator algebra. We give three additional properties, each of which characterises the standard sequential product on either a von Neumann algebra or a Euclidean Jordan algebra. These properties are (1) invariance under application of unital order isomorphisms, (2) symmetry of the sequential product with respect to a certain inner product, and (3) preservation of invertibility of the effects. To give these characterisations we first have to study convex $\sigma$-sequential effect algebras. We show that these objects correspond to unit intervals of spectral order unit spaces with a homogeneous positive cone.