Completely positive matrix numerical index on matrix regular operator spaces
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Abstract:
In the article, we compute the completely positive matrix numerical index of matrix regular operator spaces and show that they take values in the interval $[\frac {1}{2},1]$. Moreover, we show that the dual of a unital operator system has the completely positive matrix numerical index $\frac {1}{2}$ if its dimension is greater than $1$. Furthermore, both $S_p(\mathbf {H})$ and $L_p(\mathbf {M})$ have the completely positive matrix numerical index $2^{-\frac {1}{p}}$ if their dimensions are greater than $1$, where $p\in [1, + \infty )$, $\mathbf {H}$ is a Hilbert space and $\mathbf {M}$ is a finite von Neumann algebra.References
- A. J. Ellis, The duality of partially ordered normed linear spaces, J. London Math. Soc. 39 (1964), 730–744. MR 169020, DOI 10.1112/jlms/s1-39.1.730
- Edward G. Effros and Zhong-Jin Ruan, Operator spaces, London Mathematical Society Monographs. New Series, vol. 23, The Clarendon Press, Oxford University Press, New York, 2000. MR 1793753
- Kyung Hoon Han, Matrix regular operator space and operator system, J. Math. Anal. Appl. 367 (2010), no. 2, 516–521. MR 2607279, DOI 10.1016/j.jmaa.2010.01.061
- Kyung Hoon Han, Noncommutative $L_p$-space and operator system, Proc. Amer. Math. Soc. 137 (2009), no. 12, 4157–4167. MR 2538576, DOI 10.1090/S0002-9939-09-10008-4
- X.J. Huang and C.K. Ng, An abstract characterization of unital operator spaces, J. Oper. Theory, to appear.
- Richard V. Kadison, Isometries of operator algebras, Ann. of Math. (2) 54 (1951), 325–338. MR 43392, DOI 10.2307/1969534
- Anil Kumar Karn, Corrigendum to the paper: “Adjoining an order unit to a matrix ordered space” [Positivity 9 (2005), no. 2, 207–223; MR2189744], Positivity 11 (2007), no. 2, 369–374. MR 2321628, DOI 10.1007/s11117-006-2065-3
- Gerard J. Murphy, $C^*$-algebras and operator theory, Academic Press, Inc., Boston, MA, 1990. MR 1074574
- Gilles Pisier, Non-commutative vector valued $L_p$-spaces and completely $p$-summing maps, Astérisque 247 (1998), vi+131 (English, with English and French summaries). MR 1648908
- C.K. Ng, Operator subspaces of $\mathcal {L}(H)$ with induced matrix orderings, Indian. U. Math. J., to appear.
- Vern Paulsen, Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. MR 1976867
- Walter J. Schreiner, Matrix regular operator spaces, J. Funct. Anal. 152 (1998), no. 1, 136–175. MR 1600080, DOI 10.1006/jfan.1997.3160
- Walter James Schreiner, Matrix-regular orders on operator spaces, ProQuest LLC, Ann Arbor, MI, 1995. Thesis (Ph.D.)–University of Illinois at Urbana-Champaign. MR 2694086
- Wend Werner, Subspaces of $L(H)$ that are $*$-invariant, J. Funct. Anal. 193 (2002), no. 2, 207–223. MR 1929500, DOI 10.1006/jfan.2001.3943
- Wend Werner, Multipliers on matrix ordered operator spaces and some $K$-groups, J. Funct. Anal. 206 (2004), no. 2, 356–378. MR 2021851, DOI 10.1016/j.jfa.2003.05.001
Additional Information
- Xu-Jian Huang
- Affiliation: Department of Mathematics, Tianjin University of Technology, Tianjin 300384, People’s Republic of China
- Email: huangxujian86@gmail.com
- Received by editor(s): September 10, 2010
- Received by editor(s) in revised form: January 10, 2011, and March 23, 2011
- Published electronically: January 9, 2012
- Communicated by: Marius Junge
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 3161-3167
- MSC (2010): Primary 46L07, 46L52, 47L07
- DOI: https://doi.org/10.1090/S0002-9939-2012-11155-4
- MathSciNet review: 2917089