Abstract
LetE, F be exact operator spaces (for example subspaces of theC *-algebraK(H) of all the compact operators on an infinite dimensional Hilbert spaceH). We study a class of bounded linear mapsu: E →F * which we call tracially bounded. In particular, we prove that every completely bounded (in shortc.b.) mapu: E →F * factors boundedly through a Hilbert space. This is used to show that the setOS n of alln-dimensional operator spaces equipped with thec.b. version of the Banach Mazur distance is not separable ifn>2.
As an application we whow that there is more than oneC *-norm onB (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”. In the final section, we introduce and study a new tensor product forC *-albegras and for operator spaces, closely related to the preceding results.
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The second author is partially supported by the NSF.
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Junge, M., Pisier, G. Bilinear forms on exact operator spaces andB(H)⊗B(H) . Geometric and Functional Analysis 5, 329–363 (1995). https://doi.org/10.1007/BF01895670
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DOI: https://doi.org/10.1007/BF01895670