[Paper Review] Completely bounded maps into certain Hilbertian operator spaces

Gilles Pisier|arXiv (Cornell University)|Mar 13, 2004

Advanced Operator Algebra Research17 references4 citations

TL;DR

This paper establishes a factorization criterion for completely bounded maps from a C*-algebra into the complex interpolation space $(C,R)_\theta$ for $0 < \theta < 1$, showing such maps are completely bounded if and only if a certain weighted norm inequality involving states on the domain algebra holds. The key result extends the Grothendieck-type factorization for $\theta = 1/2$ to all $\theta \in (0,1)$, with constants approaching 1 as $\theta \to 0$ or $\theta \to 1$, recovering the classical cases of column and row Hilbert spaces. The proof uses generalized free Gaussian systems in non-semifinite von Neumann algebras and complex interpolation, and applies to show that $(C,R)_\theta$ does not embed completely isomorphically into the predual of any semifinite von Neumann algebra.

ABSTRACT

We prove a factorization of completely bounded maps from a $C^*$-algebra $A$ (or an exact operator space $E\subset A$) to $\ell_2$ equipped with the operator space structure of $(C,R)_θ$ ($00$ such that \[ \forall a\in A\quad \|ua\|^2\le C f(a^*a)^{1-θ}g(aa^*)^θ.\] This extends the case $θ=1/2$ treated in a recent paper with Shlyakhtenko. The constants we obtain tend to 1 when $θ o 0$ or $θ o 1$. We use analogues of "free Gaussian" families in non semifinite von Neumann algebras. As an application, we obtain that, if $0

Motivation & Objective

• To extend the factorization theorem for completely bounded maps into $\ell_2$ with the operator space structure of $(C,R)_{1/2}$ to all interpolation parameters $\theta \in (0,1)$.
• To characterize when subspaces of $R \oplus C$ embed completely isomorphically into the predual of a semifinite von Neumann algebra.
• To prove that the interpolation space $(C,R)_\theta$ does not embed completely isomorphically into the predual of any semifinite von Neumann algebra for $0 < \theta < 1$.
• To provide a new, interpolation-based proof of Junge's theorem on the complete isomorphic embedding of $OH$ into a non-commutative $L_1$-space.

Proposed method

• Use complex interpolation between the column and row Hilbert spaces $C$ and $R$ to define the operator space structure on $\ell_2$ as $(C,R)_\theta$ for $0 < \theta < 1$.
• Introduce generalized free Gaussian families in non-semifinite von Neumann algebras to model the non-commutative analogues of Gaussian variables.
• Establish a factorization criterion for completely bounded maps $u: A \to F = (C,R)_\theta$ via a norm inequality involving two states $f,g$ on the C*-algebra $A$ and a constant $C > 0$, with $\|ua\|^2 \leq C f(a^*a)^{1-\theta} g(aa^*)^\theta$.
• Apply duality and interpolation techniques to relate the dual of subspaces of $R \oplus C$ to non-commutative $L_p$-spaces.
• Use the non-commutative Khintchine inequalities and interpolation estimates to bound operator norms in $S_p[OH_n]$ and derive lower bounds on the completely bounded Banach-Mazur distance.
• Modify Junge’s original argument using complex interpolation and Shlyakhtenko’s generalized circular systems to give a unified proof of the embedding of $OH$ into non-commutative $L_1$-spaces.

Experimental results

Research questions

• RQ1 For which $\theta \in (0,1)$ does the interpolation space $(C,R)_\theta$ embed completely isomorphically into the predual of a semifinite von Neumann algebra?
• RQ2 What is the precise characterization of subspaces $S \subset R \oplus C$ whose dual $S^*$ embeds completely isomorphically into $M_*$ for some semifinite von Neumann algebra $M$?
• RQ3 Can the factorization theorem for completely bounded maps into $(C,R)_\theta$ be extended beyond $\theta = 1/2$?
• RQ4 Does $OH$ embed completely isomorphically into a non-commutative $L_1$-space, and if so, can this be proven via interpolation and free probability?
• RQ5 What is the asymptotic behavior of the completely bounded Banach-Mazur distance $d_{cb}(E, (C_n,R_n)_\theta)$ for $E$ a subspace of a non-commutative $L_q$-space?

Key findings

• Completely bounded maps $u: A \to F = (C,R)_\theta$ are characterized by the existence of states $f,g$ on $A$ and a constant $C > 0$ such that $\|ua\|^2 \leq C f(a^*a)^{1-\theta} g(aa^*)^\theta$ for all $a \in A$, with the constants $C$ approaching 1 as $\theta \to 0$ or $\theta \to 1$.
• The interpolation space $(C,R)_\theta$ does not embed completely isomorphically into the predual of any semifinite von Neumann algebra for $0 < \theta < 1$, generalizing a result of Junge.
• The only subspaces $S \subset R \oplus C$ for which $S^*$ embeds completely isomorphically into $M_*$ for some semifinite von Neumann algebra $M$ are $S = R$, $S = C$, $S = R \cap C$, and direct sums built from these.
• The completely bounded Banach-Mazur distance $d_{cb}(E, (C_n,R_n)_\theta)$ satisfies $d_{cb}(E, (C_n,R_n)_\theta) \geq (2B_q)^{-1} n^{\beta/2}$ for $q > \max\{p, p'\}$ with $p = (1-\theta)^{-1}$, $p' = \theta^{-1}$, and $\beta = \min\{p^{-1} - q^{-1}, p'^{-1} - q^{-1}\}$.
• The space $OH$ embeds completely isomorphically into a non-commutative $L_1$-space, and this result is reproven via complex interpolation and generalized free Gaussian systems, unifying earlier approaches.