Interpolation

Let $E_0$, $E_1$ be Banach spaces continuously embedded in a Hausdorff topological vector space. The pair $(E_0,E_1)$ is called a compatiple couple in the sense of interpolation theory [BL76]. Then we can define the interpolation space $E_\theta:=(E_0,E_1)_\theta$ for $0<\theta < 1$.

Pisier introduced the analogous construction for operator spaces [Pis96, §2]: Let $X_i$ $(i=0,1)$ be operator spaces continuously embedded in a Hausdorff topological vector space $V$. Then we have specific norms on $M_n(X_i)$ and continuous linear inclusions $M_n(X_i) \hookrightarrow M_n(V)$ for all $n\in{\mathbb{N}}$.⁷⁶ The interpolated operator space $X_\theta$ is defined via $M_n(X_\theta) := (M_n(X_0),M_n(X_1))_\theta$   .$$
Let $X$ be an operator space, $\H$ a Hilbert space and $V:\H\rightarrow X$ a bounded linear and injective mapping with dense range such that the mapping⁷⁷ $VV^* : \overline{X^*} \rightarrow X$ also is bounded, linear and injective with dense range. Then we have completely isometrically [Pis96, Cor. 2.4]:

$$(\overline{X^*},X)_\frac{1}{2} \stackrel{\mathrm{cb}}{=}\mathit{OH}_{\H}$$.$$$$

Examples

1. $({\mathcal{R}}_{\H},{\mathcal{C}}_{\H})_\frac{1}{2} \stackrel{\mathrm{cb}}{=}\... ...{\H} \stackrel{\mathrm{cb}}{=}(\mathit{MIN}_{\H},\mathit{MAX}_{\H})_\frac{1}{2}$
2. $({\mathcal{C}}_{\H} \otimes_h {\mathcal{R}}_{\H}, {\mathcal{R}}_{\H} \otimes_h... ...otimes_h \mathit{OH}_{\H} \stackrel{\mathrm{cb}}{=}\mathit{OH}_{\H\otimes \H}$

In this manner one also obtains operator space structures on the Schatten ideals $S_p=(S_\infty,S_1)_\frac{1}{p}$ for $1 \leq p \leq \infty$.

Footnotes

....⁷⁶

We identify $M_n(V)$ with $V^{n^2}$.

... mapping⁷⁷

As usual we identify $\H$ with its dual.