The operator Hilbert space $\mathrm{OH}$ embeds completely isometrically into a noncommutative $\mathrm{L}^1$-space

Introduction

The operator Hilbert space $OH$ introduced by Pisier in the memoir [21] is the unique operator space structure on the complex Hilbert space $\ell^2$ which is completely isometric to its conjugate dual via the canonical identification. The question of embedding $OH$ into the predual of a von Neumann algebra has a rich history. Junge [10] constructed the first completely isomorphic embedding of $OH$ into the predual of a QWEP von Neumann algebra, by means of free probability. Pisier explicitly recorded the completely isometric embedding problem as open in [23 Remark following Corollary 7.16, p. 145]. He reiterated the question in the introduction of [Pis04b], whose appendix gives a simpler proof of the completely isomorphic embedding, based on complex interpolation and Shlyakhtenko’s generalized circular systems. Pisier [24] proved that $OH$ does not even embed completely isomorphically into the predual of a semifinite von Neumann algebra. Junge [11] showed that $OH$ embeds completely isomorphically into the predual of the hyperfinite factors of type III$_\lambda$, $0 < \lambda\le1$, and Haagerup and Musat [6 Corollary 3.8 p. 620] obtained an embedding into the predual of the hyperfinite factor of type III$_1$ whose cb-isomorphism constant is at most $\sqrt{2}$. The constants in the general Khintchine inequalities underlying their construction are optimal [6 Theorem 3.10 p. 621]. Related questions appear in [28 remark p. 208 after Corollary 3.1] and in the introduction of [12]. The aim of this paper is to answer this question.

By duality, Theorem 1.1 is equivalent to the fact that the operator space OH is a complete quotient of a QWEP C$^*$-algebra, see Corollary 6.3. This is in sharp contrast with the operator spaces R + C and $R_n + C_n$ for $n \ge2$, which do not embed completely isometrically into noncommutative $L^1$-spaces according to [1], although they embed completely isomorphically [6]. By Pisier’s theorem [24], the von Neumann algebra $\mathcal{M}$ of Theorem 1.1 cannot be semifinite. Our construction takes $\mathcal{M} = (A^{**})^{\mathrm{op}}$, where $A = \prod_{\mathcal{U}} \mathrm{VN}(\mathbb{F}_{\infty})$ is the C$^*$-algebraic (norm) ultrapower of the free group factor, see Theorem 6.2 and Remark 4.3. Corollary 6.4 also gives a complete isometry into the predual of a $\sigma$-finite factor of type III.

The QWEP property gives finite-dimensional matrix approximations: for any integer $n \ge1$ and any $\delta> 0$, the operator space $\mathrm{OH}_n$ embeds into some space $S^N_1$ with cb-distortion less than $1+\delta$, by [3 Corollary 7.2], see Corollary 6.3. Combining the complete isometry with Junge’s [10 Lemmas 4.4 and 4.5] gives an exact realization of completely 1-summing norms as tensor norms in a predual. Ozawa’s compactness theorem [18] then yields convergence of the truncated models in cb-distortion for each fixed dimension, and Proposition 6.8 identifies the joint limit of their two-factor tensor norms with $\lVert T\rVert_{\pi^\circ_1}$. For the identity on $\mathrm{OH}_n$, the order $\sqrt{n(1+\log n)}$ and the numerical comparison constants are inherited from [10]. Section 7 derives the tensor distortion formula.

Approach of the paper Our proof consists of two exact computations. Let $m,n \ge1$ be integers. For any $n$-tuple $x=(x_1,\ldots,x_n)$ of matrices of $\mathrm{M}_m$, consider the completely positive maps $\Phi_x,\Phi_x^*:\mathrm{M}_m\to\mathrm{M}_m$ defined by

We first show in Theorem 3.2 that if the matrices $\Phi_x(1)$ and $\Phi_x^*(1)$ are invertible then the system of matrix equations

admits a unique solution $(P,Q)$ in positive definite matrices and that the norm of $\sum_k x_k\otimes e_k$ in $S^m_1(\mathrm{OH}_n)$ is equal to $\operatorname{Tr}(Q^{\frac{2}{3}})=\operatorname{Tr}(P^{\frac{2}{3}})$. This turns Pisier’s factorization formula [22] into a characterization by a uniquely determined pair of positive matrices. The associated map $(P,Q)\mapsto\left(\Phi_x(Q^{-\frac{1}{3}}),\Phi_x^*(P^{-\frac{1}{3}})\right)$ is a $\frac{1}{3}$-contraction for the product Thompson metric, so its iterates converge uniformly for pairs from any positive definite initial pair. The norm identity follows by constructing an optimal factorization and a norming functional, using a Perron-Frobenius argument. Thus the characterization also gives an iterative procedure for evaluating these norms, independently of the free probabilistic construction.

Next, we use a $\sigma$-system in the sense of Definition 4.1. Write $|\sigma| \mathrel{\mathop{=}^{\mathrm{def}}} \sigma((0,\infty))$. By Lemma 4.2, this system can be realized in a finite von Neumann algebra $\mathcal{A}$ with $\tau=|\sigma|\varphi$, where $\varphi is a normal faithful tracial state, and $c_k \mathrel{\mathop:}=|\sigma|^{\frac{1}{2}}c_k^{0}$, where $(c_k^{0})$ is a standard scalar-valued free circular family which is free from $\Lambda$ with respect to $\varphi$. For $D \mathrel{\mathop:}= W^{*}(\Lambda)$ and any $d \in D$, the covariance is

We put

Weighting on both sides leads to coupled covariance equations involving $\Phi_x$ and its adjoint $\Phi_x^{*}$, which can be matched with the two equations in (2). With a one-sided weighting $\Lambda^{\frac{1}{2}}c_k$, the maps $\Phi_x$ and $\Phi_x^{*}$ enter the covariance equations with different weights, and the symmetry of (2) is lost. For coefficients $x$ with $\Phi_x(1)$ and $\Phi_x^{*}(1)$ invertible, using the operator-valued Cauchy transform of the hermitization of $Z=\sum_k x_k\otimes h_k$, we show in Proposition 4.4 that the norm of $Z$ in $L^1(M_m\otimes\overline{A})$ is given by