No Access

An approximation property for operator systems

Wei Wu

School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai 200241, P. R. China

E-mail Address: wwu@math.ecnu.edu.cn

Search for more papers by this author

https://doi.org/10.1142/S021902571950005XCited by:0 (Source: Crossref)

Abstract

Motivated by an observation of Namioka and Phelps on an approximation property of order unit spaces, we introduce the $p$ -tensor product and the $m$ -tensor product of two compact matrix convex sets. We define a new approximation property for operator systems, and give a characterization using the $p$ - and $m$ -tensor products in the spirit of Grothendieck. Thus, an operator system has the operator system approximation property if and only if it is $(p, m)$ -nuclear in a natural sense.

Communicated by Michael Skeide

Keywords:

AMSC: 46L07, 47L25, 46M05, 46L89

References

1. E. M. Alfsen, Compact Convex Sets and Boundary Integrals (Springer-Verlag, Berlin, New York, 1971). Google Scholar
2. M.-D. Choi and E. G. Effros, Injectivity and operator spaces, J. Funct. Anal. 24 (1977) 156–209. Web of Science, Google Scholar
3. M.-D. Choi and E. G. Effros, Nuclear $C^{*}$ -algebras and the approximation property, Amer. J. Math. 100 (1978) 61–79. Web of Science, Google Scholar
4. E. G. Effros and Z. J. Ruan, On approximation properties for operator spaces, Internat. J. Math. 1 (1990) 163–187. Link, Google Scholar
5. E. G. Effros and Z. J. Ruan, Operator Spaces, London Mathematical Society Monographs, New Series, Vol. 23 (The Clarendon Press/Oxford Univ. Press, 2000). Google Scholar
6. E. G. Effros and C. Webster, Operator analogues of locally convex spaces, in Operator Algebras and Applications, NATO Advanced Science Institutes Series C Mathematical and Physics Sciences, Vol. 495 (Kluwer Academic Publishers, 1997), pp. 163–207. Google Scholar
7. D. R. Farenick and P. B. Morenz, $C^{*}$ -extreme points in the generalised state spaces of a $C^{*}$ -algebra, Trans. Amer. Math. Soc. 349 (1997) 1725–1748. Web of Science, Google Scholar
8. A. Grothendieck, Produits tensoriels topologiques et espaces nucleaires, Memo. Amer. Math. Soc. 16 (1955) 140. Google Scholar
9. K. H. Han and V. I. Paulsen, An approximation theorem for nuclear operator systems, J. Funct. Anal. 261 (2011) 999–1009. Web of Science, Google Scholar
10. R. V. Kadison, A representation theory for commutative topological algebra, Mem. Amer. Math. Soc. 7 (1951) 39. Google Scholar
11. A. Kavruk, V. I. Paulsen, I. G. Todorov and M. Tomforde, Tensor products of operator systems, J. Funct. Anal. 261 (2011) 267–299. Web of Science, Google Scholar
12. A. Kavruk, V. I. Paulsen, I. G. Todorov and M. Tomforde, Quotients, exactness, and nuclearity in the operator system category, Adv. Math. 235 (2013) 321–360. Web of Science, Google Scholar
13. I. Namioka and R. R. Phelps, Tensor products of compact convex sets, Pacific J. Math. 31 (1969) 469–480. Web of Science, Google Scholar
14. V. I. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge Studies in Advanced Mathematics, Vol. 78 (Cambridge Univ. Press, 2002). Google Scholar
15. C. Webster and S. Winkler, The Krein-Milman theorem in operator convexity, Trans. Amer. Math. Soc. 351 (1999) 307–322. Web of Science, Google Scholar
16. W. Wu, Non-Commutative metrics on matrix state spaces, J. Ramanujan Math. Soc. 20 (2005) 215–254. Google Scholar
17. W. Wu, Non-commutative metric topology on matrix state spaces, Proc. Amer. Math. Soc. 134 (2006) 443–453. Web of Science, Google Scholar
18. W. Wu, Quantized Gromov-Hausdorff distance, J. Funct. Anal. 238 (2006) 58–98. Web of Science, Google Scholar
19. W. Wu, An operator Arzelà-Ascoli theorem, Acta Math. Sin. (Engl. Ser.) 24 (2008) 1139–1154. Web of Science, Google Scholar
20. W. Wu, Lipschitzness of $^{*}$ -homomorphisms between $C^{*}$ -metric algebras, Sci. China Math. 54 (2011) 2473–2485. Web of Science, Google Scholar
21. W. Wu, Extremal matrix states on tensor product of $C^{*}$ -algebras, Monatsh. Math. 174 (2014) 477–491. Web of Science, Google Scholar