Tensor products of operator spaces

Abstract In this paper we lay the foundations for a systematic study of tensor products of subspaces of C ∗ -algebras. To accomplish this, various notions of duality are introduced and employed. Elementary proofs of the complete injectivity of the Haagerup norm, and of the extension theorem for completely bounded maps, are given. Pisier's gamma norms are examined and found to be special cases of the Haagerup norm. We identify the greatest operator space cross norm and show that the spatial tensor norm is the least operator space cross norm in an appropriate sense. Indeed most of the elementary theory of Banach space tensor norms generalizes to the category of operator spaces.

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