A \(p\)-theory of ordered normed spaces. (English) Zbl 1225.46014

Karn, Anil K.

Positivity 14, No. 3, 441-458 (2010).

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.

Reviewer: Mohamed Ali Toumi (Bizerte)

MSC:

46B40	Ordered normed spaces
46L07	Operator spaces and completely bounded maps

Keywords:

ordered normed spaces; order smooth \(p\)-normed spaces; order contraction; order isometry; matrix ordered space; matricially order smooth \(p\)-normed spaces; unital envelope

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