$\mathit{MIN}$ and $\mathit{MAX}$

Let $E$ be a normed space. Among all operator space norms on $E$ which coincide on the first matrix level with the given norm, there is a greatest and a smallest. The matricially normed spaces given by these are called ${\mathit{MAX}(E)}$ and ${\mathit{MIN}(E)}$. They are characterized by the following universal mapping property:¹⁴ For a matricially normed space $X$,

$$M_1(\mathit{CB}(\mathit{MAX}(E),X)) = B(E,M_1(X))$$

and

$$M_1(\mathit{CB}(X,\mathit{MIN}(E))) = B(M_1(X),E)$$

holds isometrically.

We have [Ble92a]

$$\mathit{MIN}(E)^* \stackrel{\mathrm{cb}}{=} \mathit{MAX}(E^*)$$

$$\mathit{MAX}(E)^* \stackrel{\mathrm{cb}}{=} \mathit{MIN}(E^*)$$

For ${\mathrm{dim}}(E) = \infty$,

$$\mathrm{id}_E:\mathit{MIN}(E)\rightarrow \mathit{MAX}(E)$$

is not completely bounded[Pau92, Cor. 2.13].¹⁵

Subspaces of $\mathit{MIN}$-spaces are $\mathit{MIN}$-spaces: For each isometric mapping $\varphi:E_0\to E$, the mapping $\varphi:\mathit{MIN}(E_0)\to\mathit{MIN}(E)$ is completely isometric.

Quotients of $\mathit{MAX}$-spaces are $\mathit{MAX}$-spaces: For each quotient mapping $\varphi:E\to E_0$, the mapping $\varphi:\mathit{MAX}(E)\to\mathit{MAX}(E_0)$ is a complete quotient mapping.

Footnotes

... property:¹⁴

$\mathit{MAX}$ is the left adjoint and $\mathit{MIN}$ the right adjoint of the forgetfull functor which maps an operator space $X$ to the Banach space $M_1(X)$.

...Paulsen92.¹⁵

Paulsen uses in his proof a false estimation for the projection constant of the finite dimensional Hilbert spaces; the converse estimation is correct [Woj91, p. 120], but here useless. The gap can be filled [Lam97, Thm. 2.2.15] using the famous theorem of Kadets-Snobar: The projection constant of an $n$-dimensional Banach space is less or equal than $\sqrt{n}$ [KS71].