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Tensor products of operator spaces. (English) Zbl 0786.46056
Summary: 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 cross norm in an appropriate sense. Indeed most of the elementary theory of Banach space tensor norms generalizes to the category of operator spaces.

MSC:
46M05 Tensor products in functional analysis
46L05 General theory of \(C^*\)-algebras
47A80 Tensor products of linear operators
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