The morphisms

The space $\mathit{CB}(X,Y)$ of completely bounded mappings between two homogeneous hilbertian operator spaces $X,Y$ enjoys the following properties (cf. [MP95, Prop. 1.2]):

1. $(\mathit{CB}(X,Y),\Vert\cdot\Vert _{\mathrm{cb}})$ is a Banach space.
2. $\Vert ATB\Vert _{\mathrm{cb}} \leq \Vert A\Vert \Vert T\Vert _{\mathrm{cb}} \Vert B\Vert$ for all $A$, $B \in B(\H)$, $T\in\mathit{CB}(X,Y)$.
3. $\Vert T\Vert _{\mathrm{cb}}=\Vert T\Vert$ for all $T$ with ${\rm rank}(T)=1$.

Consequently, $\mathit{CB}(X,Y)$ is a symmetrically normed ideal (s.n. ideal) in the sense of Calkin, Schatten [Sch70] and Gohberg [GK69].

The classical examples for s.n. ideals are the famous Schatten ideals:

$${S_p} := \left\{ T \in B(\H) \left\vert \mbox{ the sequence of \... ... T \mbox{ is in } \ell_p \right. \right\} \quad (1 \leq p < \infty) \mbox{.}$$

Many, but not all s.n. ideals can be represented as spaces of completely bounded maps between suitable homogeneous hilbertian operator spaces.

The first result in this direction was

$$\mathit{CB}({\mathcal{R}}_{\H},{\mathcal{C}}_{\H})=S_2(\H)=\mathit{HS}(\H)$$

isometrically [ER91, Cor. 4.5].

We have the following characterizations isometrically or only isomorphically ($\simeq$) [Mat94], [MP95], [Lam97]:

$\mathit{CB}(\downarrow,\rightarrow)$	$\mathit{MIN}_{\H}$	${\mathcal{C}}_{\H}$	$\mathit{OH}_{\H}$	${\mathcal{R}}_{\H}$	$\mathit{MAX}_{\H}$
$\mathit{MIN}_{\H}$	$B(\H)$	$\mathit{HS}(\H)$	$\simeq \mathit{HS}(\H)$	$\mathit{HS}(\H)$	$\simeq N(\H)$
${\mathcal{C}}_{\H}$	$B(\H)$	$B(\H)$	$S_4(\H)$	$\mathit{HS}(\H)$	$\mathit{HS}(\H)$
$OH_{\H}$	$B(\H)$	$S_4(\H)$	$B(\H)$	$S_4(\H)$	$\simeq \mathit{HS}(\H)$
${\mathcal{R}}_{\H}$	$B(\H)$	$\mathit{HS}(\H)$	$S_4(\H)$	$B(\H)$	$\mathit{HS}(\H)$
$\mathit{MAX}_{\H}$	$B(\H)$	$B(\H)$	$B(\H)$	$B(\H)$	$B(\H)$

As a unique completely isometric isomorphism, we get $\mathit{CB}({\mathcal{C}}_\H) \stackrel{\mathrm{cb}}{=}B(\H)$ (cf. [Ble95, Thm. 3.4]). The result

$$\mathit{CB}(\mathit{MIN}_{\H},\mathit{MAX}_{\H}) \simeq N(\H)$$

is of special interest. Here, we have a new quite natural norm on the nuclear operators, which is not equal to the canonical one.

Even in the finite dimensional case, we only know

$$\frac{n}{2} \leq \Vert\mathrm{id}:\mathit{MIN}(\ell_2^n) \rightarrow \mathit{MAX}(\ell_2^n) \Vert _{\mathrm{cb}} \leq \frac{n}{\sqrt{2}}$$

[Pau92, Thm. 2.16]. To compute the exact constant is still an open problem. Paulsen conjectured that the upper bound is sharp [Pau92, p. 121].

Let $E$ be a Banach space and $\H$ a Hilbert space. An operator $T \in B(E,\H)$ is completely bounded from $\mathit{MIN}(E)$ to ${\mathcal{C}}_\H$ , if and only if $T$ is 2-summing [Pie67] from $E$ to $\H$ (cf. [ER91, Thm. 5.7]):

$$M_1(\mathit{CB}(\mathit{MIN}(E),{\mathcal{C}}_\H)) = {\Pi_2}(E,\H)$$

with $\Vert T\Vert _\mathrm{cb}=$ ${\pi_2}(T)$.