[ m at h . FA ] 2 4 A ug 1 99 3 Bilinear forms on exact operator spaces and B ( H ) ⊗ B ( H ) . by Marius Junge and

Let E, F be exact operator spaces (for example subspaces of the C *-algebra K(H) of all the compact operators on an infinite dimensional Hilbert space H). We study a class of bounded linear maps u: E → F * which we call tracially bounded. In particular, we prove that every completely bounded (in short c.b.) map u: E → F * factors boundedly through a Hilbert space. This is used to show that the set OS n of all n-dimensional operator spaces equipped with the c.b. version of the Banach Mazur distance is not separable if n > 2. As an application we show that there is more than one C *-norm on B(H) ⊗ B(H), or equivalently that which answers a long standing open question. Finally we show that every " maximal " operator space (in the sense of Blecher-Paulsen) is not exact in the infinite dimensional case, and in the finite dimensional case, we give a lower bound for the " exactness constant " .