Injectivity and operator spaces |
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Authors: | Man-Duen Choi Edward G. Effros |
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Affiliation: | Department of Mathematics, University of Toronto, Toronto, M5S 1A1, Canada;Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104 USA |
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Abstract: | Let B(H) be the bounded operators on a Hilbert space H. A linear subspace R ? B(H) is said to be an operator system if 1 ?R and R is self-adjoint. Consider the category of operator systems and completely positive linear maps. R ∈ is said to be injective if given A ? B, A, B ∈ , each map A → R extends to B. Then each injective operator system is isomorphic to a conditionally complete C1-algebra. Injective von Neumann algebras R are characterized by any one of the following: (1) a relative interpolation property, (2) a finite “projectivity” property, (3) letting Mm = B(Cm), each map R → N ? Mm has approximate factorizations R → Mn → N, (4) letting K be the orthogonal complement of an operator system N ? Mm, each map has approximate factorizations . Analogous characterizations are found for certain classes of C1-algebras. |
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