The mappings

A linear mapping $\Phi$ between vector spaces $X$ and $Y$ induces a linear mapping $\Phi^{(n)}=\mathrm{id}_{M_n} \otimes \Phi$ ,

$$\Phi^{(n)}:M_n(X) \to M_n(Y)$$

$$\left[x_{ij}\right] \mapsto \left[\Phi(x_{ij})\right]$$

the $n$th amplification of $\Phi$.

For matricially normed $X$ and $Y$, one defines

$$\Vert\Phi\Vert _{\mathrm{cb}} :=\sup\left\{\left. \Vert\Phi^{(n)}\Vert \; \right\vert \; n\in {\mathbb{N}}\right\}$$   .

$\Phi$ is called completely bounded if $\Vert\Phi\Vert _{\mathrm{cb}}<\infty$ and completely contractive if $\Vert\Phi\Vert _{\mathrm{cb}}\leqslant 1$.

Among the complete contractions, the complete isometries and the complete quotient mappings play a special role. $\Phi$ is called completely isometric if all $\Phi^{(n)}$ are isometric,⁵and a complete quotient mapping if all $\Phi^{(n)}$ are quotient mappings.⁶

The set of all completely bounded mappings from $X$ to $Y$ is denoted by $\mathit{CB}(X,Y)$ [Pau86, Chap. 7].

An operator space $X$ is called homogeneous if each bounded operator $\Phi \in B(M_1(X))$ is completely bounded with the same norm: $\Phi\in CB(X)$, and $\Vert\Phi\Vert _{\mathrm{cb}}=\Vert\Phi\Vert$ [Pis96].

Footnotes

... isometric,⁵

I. e.: $\Vert x\Vert=\Vert\Phi^{(n)}(x)\Vert$ for all $n\in{\mathbb{N}}$, $x\in M_n(X)$.

... mappings.⁶

I. e.: $\Vert y\Vert=\inf\{\Vert x\Vert\;\vert\;x\in {\Phi^{(n)}}^{-1}(y)\}$ for alle $n\in{\mathbb{N}}$ and $y\in M_n(Y)$, or equivalently $\Phi^{(n)}(\mathrm{Ball}^\circ M_n(X))=\mathrm{Ball}^\circ M_n(Y)$ for all $n\in{\mathbb{N}}$, where $\mathrm{Ball}^\circ M_n(X) = \{x\in M_n(X)\;\vert\;\Vert x\Vert< 1\}$.