Injective matricial Hilbert spaces

Abstract

Injective matricial operator spaces have been classified up to Banach space isomorphism in [20]. The result is that every such space is isomorphic to l∞, l2, B(l2), or a direct sum of such spaces. A more natural project, given the matricial nature of the definitions involved, would be the classification of such spaces up to completely bounded isomorphism. This was done for injective von Neumann algebras in [6] and for injective operator systems (i.e. unital injective operator spaces) in [19]. It turns out that the spaces l∞ and B(l2) are in a natural way uniquely characterized up to completely bounded isomorphism. However, as shown in [20], a problem arises in the case of l2. For there are two injective operator spaces which are each isometrically isomorphic to l2 but not completely boundedly isomorphic to each other. We shall resolve this problem by showing that these are the only two possibilities, in the sense that any injective operator space which is isometric to l2 is completely isometric to one of them. (See Corollary 3 below.) The Hilbert spaces in von Neumann algebras investigated in [17], [13] turn out to be injective matricial operator spaces and are therefore completely isometric to one of our two examples. Another Hilbert space in B(l2) which has been much studied in operator theory, complex analysis and physics is the Cartan factor of type IV [10]. This is the complex linear span of a spin system and generates the Fermion C*-algebra ([3], §5·2). We show that a Cartan factor of type IV is not even completely boundedly isomorphic to an injective matricial operator space. One curious property of all the aforementioned Hilbert spaces is that every bounded operator on them is actually completely bounded, a fact that is crucial in our proofs.