Three characterisations of the sequential product

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.