BILINEAR-FORMS ON EXACT OPERATOR-SPACES AND B(H)CIRCLE-TIMES-B(H)
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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 OSn 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 whow that there is more than one C*-norm on B (H) B (H), or equivalently that {Mathematical expression} 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". In the final section, we introduce and study a new tensor product for C*-albegras and for operator spaces, closely related to the preceding results. 1995 Birkhuser Verlag.