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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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[1] Arveson W.B.: Subalgebras of C*-algebras. Acta Math. 123, 141–224 (1969) · Zbl 0194.15701 · doi:10.1007/BF02392388
[2] Blecher, D.P., Neal, M.: Open partial isometries and positivity in operator spaces (preprint, 2006) · Zbl 1130.46034
[3] Choi M.D., Effros E.G.: Injectivity and operator spaces. J. Funct. Anal. 24, 156–209 (1977) · Zbl 0341.46049 · doi:10.1016/0022-1236(77)90052-0
[4] Jameson, G.J.O.: Ordered linear spaces. Lecture Notes in Mathematics, vol. 141. Springer, New York (1970) · Zbl 0196.13401
[5] Karn A.K.: Adjoining an order unit in a matrix ordered space. Positivity 9, 207–223 (2005) · Zbl 1116.46046 · doi:10.1007/s11117-003-2778-5
[6] Karn A.K.: Corrigendum to the paper ”Adjoining an order unit in a matrix ordered space”. Positivity 11, 369–374 (2007) · Zbl 1116.46047 · doi:10.1007/s11117-006-2065-3
[7] Karn, A.K.: Order embedding of matrix ordered spaces. (Communicated for publication)
[8] Karn A.K., Vasudevan R.: Matrix norms in matrix ordered spaces. Glasnik Mathematici 32(52), 87–97 (1997) · Zbl 0885.46015
[9] Karn A.K., Vasudevan R.: Approximate matrix order unit spaces. Yokohama Math. J. 44, 73–91 (1997) · Zbl 0902.46030
[10] Karn A.K., Vasudevan R.: Matrix duality for matrix ordered spaces. Yokohama Math. J. 45, 1–18 (1998) · Zbl 0944.46011
[11] Karn A.K., Vasudevan R.: Characterizations of matricially Riesz normed spaces. Yokohama Math. J. 47, 143–153 (2000) · Zbl 0965.46002
[12] Ruan Z.J.: Subspaces of C*-algebras. J. Funct. Anal. 76, 217–230 (1988) · Zbl 0646.46055 · doi:10.1016/0022-1236(88)90057-2
[13] Schreiner W.J.: Matrix regular operator spaces. J. Funct. Anal. 152, 136–175 (1998) · Zbl 0898.46017 · doi:10.1006/jfan.1997.3160
[14] Werner W.: Subspaces of L(H) that are *-invariant. J. Funct. Anal. 193, 207–223 (2002) · Zbl 1020.46015 · doi:10.1006/jfan.2001.3943
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