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arXiv · 1803.08453

Three characterisations of the sequential product

John van de Wetering
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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 $σ$-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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John van de Wetering. 2018-03-13. Three characterisations of the sequential product. https://doi.org/10.1063/1.5031089

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