Representation of certain homogeneous Hilbertian operator spaces and applications

• 成果类型: Article
• 署名作者: Junge, Marius; Xu, Quanhua
• 署名单位: Universite Marie et Louis Pasteur; University of Illinois System; University of Illinois Urbana-Champaign
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-009-0210-x
• 发表日期: 2010
• 页码: 75-118
• 关键词: martingale difference-sequences; banach-spaces

摘要

Following Grothendieck's characterization of Hilbert spaces we consider operator spaces F such that both F and F* completely embed into the dual of a C*-algebra. Due to Haagerup/Musat's improved version of Pisier/Shlyakhtenko's Grothendieck inequality for operator spaces, these spaces are quotients of subspaces of the direct sum C circle plus R of the column and row spaces (the corresponding class being denoted by QS(C circle plus R)). We first prove a representation theorem for homogeneous F is an element of QS(C circle plus R) starting from the fundamental sequences Phi(c)(n) = parallel to Sigma(n)(k=1) e(k1) circle times e(k)parallel to(2)(C circle times minF) and Phi(r)(n) = parallel to Sigma(n)(k=1) e(1k) circle times e(k)parallel to(2)(R circle times minF) given by an orthonormal basis (e(k)) of F. Under a mild regularity assumption on these sequences we show that they completely determine the operator space structure of F and find a canonical representation of this important class of homogeneous Hilbertian operator spaces in terms of weighted row and column spaces. This canonical representation allows us to get an explicit formula for the exactness constant of an n-dimensional subspace F-n of F: ex(F-n) similar to [n/Phi(c)(n)Phi(r)(Phi(c)(n)/Phi(r)(n) + n/Phi(r)(n)Phi(c)(Phi(r)(n)/Phi(c)(n))](1/2). In the same way, the projection (=injectivity) constant of F-n is explicitly expressed in terms of Phi(c) and Phi(r) too. Orlicz space techniques play a crucial role in our arguments. They also permit us to determine the completely 1-summing maps in Effros and Ruan's sense between two homogeneous spaces E and F in QS(C circle plus R). The resulting space Pi(0)(1) (E, F) isomorphically coincides with a Schatten-Orlicz class S-phi.. Moreover, the underlying Orlicz function phi is uniquely determined by the fundamental sequences of E and F. In particular, applying these results to the column subspace C-p of the Schatten p-class, we find the projection and exactness constants of C-p(n), and determine the completely 1-summing maps from C-p to C-q for any 1 <= p, q <= infinity.