Isomorphisms of Injective Operator Spaces and Jordan Triple Systems

Injective matricial Hilbert spaces

“…There is no real difficulty in extending this to the non-separable case. This is in contrast to the results of [19,20] where the completely bounded isomorphism class of I m is only known for operator spaces acting on a separable Hilbert space.…”

“…An injective (matricial) operator space E is the range of a completely contractive projection on B(H), the algebra of bounded operators on a complex Hilbert space H. Ruan [22] has shown that E is completely isometric to pAq where A is an injective C*-algebra and p,q are projections in A such that p + q = 1. This result was crucial to the classification obtained in [20] and also in what follows.…”

“…Injective matricial operator spaces have been classified up to Banach space isomorphism in [20]. The result is that every such space is isomorphic to Z°°,I 2 , B(l 2 ), or a direct sum of such spaces.…”

“…It turns out that the spaces l x and B(l 2 ) are in a natural way uniquely characterized up to completely bounded isomorphism. However, as shown in [20], a problem arises in the case of I 2 . For there are two injective operator spaces which are each isometrically isomorphic to I 2 but not completely boundedly isomorphic to each other.…”

A continuity theorem for Stinespring's dilation

“…We refer to Chapter XVI in Takesaki's textbook [21] for this and further equivalent conditions for injectivity of von Neumann algebras. A characterization of injective C * -algebras has been given by Robertson et al [15,16]. For cp maps with non-injective range, we only have a lower bound on β(T 1 , T 2 ), though we could always apply Theorem 1 to the concatenated maps σ • T 1 with some faithful embedding σ .…”