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A $$p$$-theory of ordered normed spaces. (English) Zbl 1225.46014
Author’s abstract: We propose a pair of axioms ($$O.p.1$$) and ($$O.p.2$$) for $$1 \leq p \leq \infty$$ and initiate a study of a (matrix) ordered space with a (matrix) norm in which the (matrix) norm is related to the (matrix) order. We call such a space a (matricially) order smooth $$p$$-normed space. The advantage of studying these spaces over $$L^{p}$$-matricially Riesz normed spaces is that every matricially order smooth $$\infty$$-normed space can be order embedded in some $$C^*$$-algebra. We also study the adjoining of an order unit to a (matricially) order smooth $$\infty$$-normed space. As a consequence, we sharpen Arveson’s extension theorem of completely positive maps. Another combination of these axioms yields an order theoretic characterization of the set of real numbers amongst ordered normed linear spaces.

##### MSC:
 46B40 Ordered normed spaces 46L07 Operator spaces and completely bounded maps
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##### References:
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