Sequential Product Spaces are Jordan Algebras

TL;DR

This paper proves that finite-dimensional sequential product spaces are equivalent to Euclidean Jordan algebras, and explores their implications for quantum theory and the structure of operator algebras.

Contribution

It establishes a characterization of sequential product spaces as Jordan algebras, connecting order unit spaces with quantum and operator algebra structures.

Findings

Finite-dimensional sequential product spaces are homogeneous and self-dual.

Such spaces correspond to Euclidean Jordan algebras via Koecher-Vinberg theorem.

In infinite dimensions, they relate to JB-algebras under certain conditions.

Abstract

We show that finite-dimensional order unit spaces equipped with a continuous sequential product as defined by Gudder and Greechie are homogeneous and self-dual. As a consequence of the Koecher-Vinberg theorem these spaces therefore correspond to Euclidean Jordan algebras. We remark on the significance of this result in the context of reconstructions of quantum theory. In particular, we show that sequential product spaces must be C*-algebras when their vector space tensor product is also a sequential product space (in the parlance of operational theories, when the space `allows a local composite'). We also show that sequential product spaces in infinite dimension correspond to JB-algebras when a few additional conditions are satisfied. Finally, we remark on how changing the axioms of the sequential product might lead to a new characterisation of homogeneous cones.