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The base-normed space of a unital group. (English) Zbl 1077.81007
Summary: One of the more elegant approaches to the mathematical foundations of the experimental sciences is the linear-duality formalism featuring an order-unit space \(U\) in order duality with a base-normed space \(V\). The unit interval \(E\) in \(U\) is the set of effects, and the cone base \(\Omega \) in \(V\) is the set of states. For various reasons, some of which we indicate, it is useful to replace the order-unit space \(U\) by a partially ordered abelian group \(G\) with order unit. One can still associate a base-normed space \(V(G)\) with \(G\), and much of the articulation between \(U\) and \(V\) is still available in this more general context.
MSC:
81P10 Logical foundations of quantum mechanics; quantum logic (quantum-theoretic aspects)
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
46B40 Ordered normed spaces
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
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