Self-duality for the Haagerup tensor product and Hilbert space factorizations

Abstract

D. Blecher and V. Paulsen showed that the Haagerup tensor product V ⊗ h W for operator spaces V and W preserves inclusions. It is proved to also preserve complete quotient maps, and to be self-dual in the sense that it induces the Haagerup norm on the algebraic tensor product V ∗ ⊗ W ∗ . The full operator dual space (V ⊗ h W) ∗ is computed. It coincides with the natural operator space \\ ̃ gG 2(V, W ∗) of maps ϑ: V → W ∗ which have completely bounded factorizations through Hilbert spaces (with vectors identified with row matrices). More generally, one has the natural complete isometry \\ ̃ gG 2(V ⊗ h W, X) ≊ \\ ̃ gG 2(V, \\ ̃ gG 2(W, X)) . Given Hilbert spaces H and K with vectors regarded as column matrices, it is shown that one may identify the operator spaces B( H, K) and CB( H, K).