Completely positive matrix numerical index on matrix regular operator spaces

Author:
Xu-Jian Huang

Journal:
Proc. Amer. Math. Soc. **140** (2012), 3161-3167

MSC (2010):
Primary 46L07, 46L52, 47L07

Published electronically:
January 9, 2012

MathSciNet review:
2917089

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Abstract | References | Similar Articles | Additional Information

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 . Moreover, we show that the dual of a unital operator system has the completely positive matrix numerical index if its dimension is greater than . Furthermore, both and have the completely positive matrix numerical index if their dimensions are greater than , where , is a Hilbert space and is a finite von Neumann algebra.

**1.**A. J. Ellis,*The duality of partially ordered normed linear spaces*, J. London Math. Soc.**39**(1964), 730–744. MR**0169020****2.**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****3.**Kyung Hoon Han,*Matrix regular operator space and operator system*, J. Math. Anal. Appl.**367**(2010), no. 2, 516–521. MR**2607279**, 10.1016/j.jmaa.2010.01.061**4.**Kyung Hoon Han,*Noncommutative 𝐿_{𝑝}-space and operator system*, Proc. Amer. Math. Soc.**137**(2009), no. 12, 4157–4167. MR**2538576**, 10.1090/S0002-9939-09-10008-4**5.**X.J. Huang and C.K. Ng, An abstract characterization of unital operator spaces, J. Oper. Theory, to appear.**6.**Richard V. Kadison,*Isometries of operator algebras*, Ann. Of Math. (2)**54**(1951), 325–338. MR**0043392****7.**Anil Kumar Karn,*Corrigendum to the paper: “Adjoining an order unit to a matrix ordered space” [Positivity 9 (2005), no. 2, 207–223; MR 2189744]*, Positivity**11**(2007), no. 2, 369–374. MR**2321628**, 10.1007/s11117-006-2065-3**8.**Gerard J. Murphy,*𝐶*-algebras and operator theory*, Academic Press, Inc., Boston, MA, 1990. MR**1074574****9.**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****10.**C.K. Ng, Operator subspaces of with induced matrix orderings, Indian. U. Math. J., to appear.**11.**Vern Paulsen,*Completely bounded maps and operator algebras*, Cambridge Studies in Advanced Mathematics, vol. 78, Cambridge University Press, Cambridge, 2002. MR**1976867****12.**Walter J. Schreiner,*Matrix regular operator spaces*, J. Funct. Anal.**152**(1998), no. 1, 136–175. MR**1600080**, 10.1006/jfan.1997.3160**13.**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****14.**Wend Werner,*Subspaces of 𝐿(𝐻) that are *-invariant*, J. Funct. Anal.**193**(2002), no. 2, 207–223. MR**1929500**, 10.1006/jfan.2001.3943**15.**Wend Werner,*Multipliers on matrix ordered operator spaces and some 𝐾-groups*, J. Funct. Anal.**206**(2004), no. 2, 356–378. MR**2021851**, 10.1016/j.jfa.2003.05.001

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Additional Information

**Xu-Jian Huang**

Affiliation:
Department of Mathematics, Tianjin University of Technology, Tianjin 300384, People’s Republic of China

Email:
huangxujian86@gmail.com

DOI:
https://doi.org/10.1090/S0002-9939-2012-11155-4

Keywords:
Completely positive matrix numerical index,
matrix regular operator space.

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

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.