THE OPERATOR HILBERT SPACE OH AND TYPE III VON NEUMANN ALGEBRAS

Abstract

A proof is given to show 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, which 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 [10]) embeds completely isomorphically 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 paper, we show that no such embedding can exist when M is semi-finite. The results that we have just stated all belong to the currently very active field of ‘operator spaces’, for which we refer the reader to the monographs [2, 11]. We merely recall a few basic facts, relevant to the present paper. An operator space is a Banach space, given together with an isometric embedding E ⊂ B(H )i nto 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).