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 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 the 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 σ-sequential effect algebras. We show that these objects correspond to unit intervals of spectral order unit spaces with a homogeneous positive cone.
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Research Article| August 21 2018
Three characterisations of the sequential product
John van de Wetering; Three characterisations of the sequential product. J. Math. Phys. 1 August 2018; 59 (8): 082202. https://doi.org/10.1063/1.5031089
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