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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.

46B40 Ordered normed spaces
46L07 Operator spaces and completely bounded maps
Full Text: DOI
[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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