On maximal tensor products and quotient maps of operator systems. (English) Zbl 1242.46067
Summary: We introduce quotient maps in the category of operator systems and show that the maximal tensor product is projective with respect to them, whereas the maximal tensor product is not injective, which makes the (el,max)-nuclearity distinguish a class in the category of operator systems. We generalize Lance’s characterization of \(C^{*}\)-algebras with the WEP by showing that (el,max)-nuclearity is equivalent to the weak expectation property. Applying Werner’s unitization to the dual spaces of operator systems, we consider a class of completely positive maps associated with the maximal tensor product and establish the duality between quotient maps and complete order embeddings.
MSC:
| 46L07 | Operator spaces and completely bounded maps |
| 47L25 | Operator spaces (= matricially normed spaces) |
| 46L35 | Classifications of \(C^*\)-algebras |
References:
| [1] | Alfsen, E. M., Compact Convex Sets and Boundary Integrals, Ergeb. Math. Grenzgeb., vol. 57 (1971), Springer-Verlag: Springer-Verlag New York-Heidelberg · Zbl 0209.42601 |
| [2] | Choi, M.-D.; Effros, E. G., Injectivity and operator spaces, J. Funct. Anal., 24, 156-209 (1977) · Zbl 0341.46049 |
| [3] | Conway, J. B., A Course in Functional Analysis, Grad. Texts in Math., vol. 96 (1990), Springer-Verlag: Springer-Verlag New York · Zbl 0706.46003 |
| [4] | Han, K. H., Matrix regular operator space and operator system, J. Math. Anal. Appl., 367, 2, 516-521 (2010) · Zbl 1198.46045 |
| [5] | Han, K. H.; Paulsen, V. I., An approximation theorem for nuclear operator systems, J. Funct. Anal., 261, 999-1009 (2011) · Zbl 1223.46053 |
| [6] | Kadison, R. V., A representation theory for commutative topological algebra, Mem. Amer. Math. Soc., 1951, 7 (1951) · Zbl 0042.34801 |
| [7] | Karn, A. K., Corrigendum to the paper: “Adjoining an order unit to a matrix ordered space”, Positivity, 11, 2, 369-374 (2007) · Zbl 1116.46047 |
| [8] | Kavruk, A.; Paulsen, V. I.; Todorov, I. G.; Tomforde, M., Tensor products of operator systems, J. Funct. Anal., 261, 267-299 (2011) · Zbl 1235.46051 |
| [9] | Kavruk, A.; Paulsen, V. I.; Todorov, I. G.; Tomforde, M., Quotients, exactness and WEP in the operator systems category, preprint · Zbl 1325.46060 |
| [10] | Kirchberg, E.; Wassermann, S., \(C^\ast \)-algebras generated by operator systems, J. Funct. Anal., 155, 2, 324-351 (1998) · Zbl 0940.46038 |
| [11] | Lance, C., On nuclear \(C^\ast \)-algebras, J. Funct. Anal., 12, 157-176 (1973) · Zbl 0252.46065 |
| [12] | Lance, C., Tensor products and nuclear \(C^\ast \)-algebras, (Operator Algebras and Applications, Part I. Operator Algebras and Applications, Part I, Kingston, Ont., 1980. Operator Algebras and Applications, Part I. Operator Algebras and Applications, Part I, Kingston, Ont., 1980, Proc. Sympos. Pure Math., vol. 38 (1982), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 379-399 · Zbl 0498.46044 |
| [13] | Paulsen, V. I.; Todorov, I. G.; Tomforde, M., Operator system structures on ordered spaces, Proc. Lond. Math. Soc. (3), 102, 25-49 (2011) · Zbl 1213.46050 |
| [14] | Paulsen, V. I.; Tomforde, M., Vector spaces with an order unit, Indiana Univ. Math. J., 58, 3, 1319-1359 (2009) · Zbl 1211.46016 |
| [15] | Paulsen, V. I., Completely Bounded Maps and Operator Algebras, Cambridge Stud. Adv. Math., vol. 78 (2002) · Zbl 1029.47003 |
| [16] | Werner, W., Subspaces of \(L(H)\) that are ⁎-invariant, J. Funct. Anal., 193, 207-223 (2002) · Zbl 1020.46015 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
