Representation of certain homogeneous Hilbertian operator spaces and applications

Abstract

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 ⊕ R of the column and row spaces (the corresponding class being denoted by QS(C ⊕ R)). We first prove a representation theorem for homogeneous F∈QS(C ⊕ R) starting from the fundamental sequences

$$\Phi _{c}(n)=\Bigg\|\sum_{k=1}^ne_{k1}\otimes e_k\Bigg\|_{C\otimes _{\min}F}^2\quad\mbox{and}\quad \Phi _{r}(n)=\Bigg\|\sum_{k=1}^ne_{1k}\otimes e_k\Bigg\|_{R\otimes _{\min}F}^2$$

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:

$$\mathit{ex}(F_n)\sim\biggl[\frac{n}{ \Phi _{c}(n)}\Phi _{r}\bigg(\frac{ \Phi _{c}(n)}{\Phi _{r}(n)}\bigg)+\frac{n}{ \Phi _{r}(n)}\Phi _{c}\bigg(\frac{ \Phi _{r}(n)}{\Phi _{c}(n)}\bigg)\biggr]^{1/2}.$$

In the same way, the projection (=injectivity) constant of F _n is explicitly expressed in terms of Φ _c and Φ _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 ⊕ R). The resulting space Π ^o₁ (E, F) isomorphically coincides with a Schatten-Orlicz class S _φ . Moreover, the underlying Orlicz function φ 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 ⁿ _p , and determine the completely 1-summing maps from C _p to C _q for any 1≤p, q≤∞.