The Operator Hilbert Space OH and TYPE III von Neumann Algebras

We prove that the operator Hilbert space OH does not embed completely isomorphically into the predual of a semi-finite von Neumann algebra. This complements Junge’s recent result that it admits such an embedding in the non semi-finite case. ——————– In remarkable recent work [5], Marius Junge proves that the operator Hilbert space OH (from [8], see also [9]) embeds completely isomorphically (c.i. in short) into the predual M∗ of a von Neumann algebra M which is of type III; thus this algebra M is not semi-finite. In this note, we show that no such embedding can exist when M is semi-finite. The results we just stated all belong to the currently very active field of “operator spaces” for which we refer the reader to the monographs [2, 10]. We merely recall a few basic facts relevant for the present note. An operator space is a Banach space given together with an isometric embedding E ⊂ B(H) into the algebra B(H) of all bounded operators on a Hilbert space H . Using this embedding, we equip the space Mn(E) (consisting of the n× n matrices with entries in E) with the norm induced by the space Mn(B(H)), naturally identified isometrically with B(H ⊕ · · · ⊕ H). Let F ⊂ B(K) be another operator space. In operator space theory, the morphisms are the completely bounded (c.b. in short) linear maps: A linear map u : E → F is called c.b. if the mappings un : Mn(E) → Mn(F ) defined by [xij ] → [u(xij)] are uniformly bounded when n ranges over all integers ≥ 1, and the cb-norm is defined as ‖u‖cb = supn ‖un‖. The resulting normed space of all c.b. maps u : E → F equipped with the cb-norm is denoted MSC 2000: 46L07, 46L54, 47L25, 47L50 Partially supported by NSF and Texas Advanced Research Program 010366-163