Operator space projective tensor product: Embedding into second dual and\n ideal structure

2011/06/14 by Ranjana Jain, Ajay Kumar, Jain, Ranjana +1
Mathematics · Physics and Astronomy · #46L06 #46L07 #47L25 #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Noncommutative and Quantum Gravity Theories #Operator Algebras (math.OA)

paper · pdf · doi:10.48550/arxiv.1106.2644

openalex publication_date 2011/06/14 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

We prove that for operator spaces V and W, the operator space\nV^**\⊗_h W^** can be completely isometrically embedded into\n(V\⊗_h W)^**, \⊗_h being the Haagerup tensor product. It is\nalso shown that, for exact operator spaces V and W, a jointly completely\nbounded bilinear form on V\× W can be extended uniquely to a separately\nw^*-continuous jointly completely bounded bilinear form on V^**\×\nW^**. This paves the way to obtain a canonical embedding of\nV^**\\⊗ W^** into (V\\⊗ W)^** with a continuous\ninverse, where \\⊗ is the operator space projective tensor product.\nFurther, for C^*-algebras A and B, we study the (closed) ideal structure\nof A\\⊗B, which, in particular, determines the lattice of closed\nideals of B(H)\\⊗ B(H) completely.\n

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