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

Edward G. Effros, Zhong Jin Ruan

Research output: Contribution to journalArticlepeer-review


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 \ ̃gG2(V, W*) of maps θ{symbol}: V → W* which have completely bounded factorizations through Hilbert spaces (with vectors identified with row matrices). More generally, one has the natural complete isometry \ ̃gG2(V ⊗h W, X) ≊ \ ̃gG2(V, \ ̃gG2(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).

Original languageEnglish (US)
Pages (from-to)257-284
Number of pages28
JournalJournal of Functional Analysis
Issue number2
StatePublished - Sep 1991

ASJC Scopus subject areas

  • Analysis

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