  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.
