Completely integral mappings

Completely integral mappings are defined by the aid of completely nuclear mappings . A mapping $\varphi:X\to Y$ is said to be completely integral, if there are a constant $c>0$ and a net of finite rank maps $\varphi_\alpha \in \mathit{CN}(X,Y)$ with $\nu(\varphi_\alpha)\leq c$ converging to $\varphi$ in the point norm topology ⁷⁰.

The set of all these mappings forms the space $\mathit{CI}(X,Y)$ of the completely integral mappings.

The infimum of all those constants $c$ satisfying the above condition is actually attained and denoted by $\iota(\varphi)$. $\iota(\cdot)$ is a norm turning $\mathit{CI}(X,Y)$ into a Banach space. The unit sphere of $\mathit{CI}(X,Y)$ is merely the point norm closure of the unit sphere of $\mathit{CN}(X,Y)$.

One obtains the canonical operator space structure by defining the unit sphere of $M_n(\mathit{CI}(X,Y))$ as the point norm closure of unit sphere of $M_n(\mathit{CN}(X,Y))$.

By definition we have $\iota(\varphi)\leq\nu(\varphi)$; for finite dimensional $X$ we have moreover [EJR98, Lemma 4.1]

$$\mathit{CI}(X,Y)\stackrel{\mathrm{cb}}{=}\mathit{CN}(X,Y)$$.$$$$

Integral ⁷¹mappings are completely integral [ER94, 3.10].

The canonical embedding

$$\mathit{CI}(X,Y)\hookrightarrow (X \stackrel{\scriptscriptstyle \vee}{\otimes}Y^*)^*$$

is a complete isometry [EJR98, Cor. 4.3]. One has moreover that [EJR98, Cor. 4.6] $\varphi$ is completely integral, if and only if there is a factorization of the form:

$$B({\cal H}) \stackrel{M(\omega)}{\rightarrow } B({\cal K})^*$$

$$\uparrow r \downarrow s$$

$$X\quad \stackrel{\varphi}{\rightarrow } Y\hookrightarrow Y^{**}$$

with weak$^*$-continuous $s$ . The mapping $M(\omega):B({\cal H})\to B({\cal K})^*$ is for two elements $a\in B({\cal H})$, $b\in B({\cal K})$ given by $(M(\omega)(a))(b)=\omega(a\otimes b)$. We have for the norm $\iota(\varphi)= 1$, if there is a factorization with $\Vert r\Vert _{cb}\Vert\omega \Vert\Vert s\Vert _{cb}= 1$ (note that generally $\Vert M(\omega)\Vert _{cb}\neq\Vert\omega\Vert$).

The completely integral mappings enjoy also the $\mathit{CB}$-ideal property . Contrasting the situation of completely nuclear mappings they are local . In general one only has $\iota(\varphi)\leq\iota(\varphi^*)$ [EJR98].

Footnotes

... topology⁷⁰

$\varphi_\alpha\to \varphi$ in the point norm topology, if $\Vert\varphi_\alpha(x)-\varphi(x)\Vert\to 0$ for all $x \in X$.

...Integral ⁷¹

The unit ball of the integral mappings of the Banach space theory is just the point norm closure of the unit ball of the nuclear mappings. One should note that the formulas $\iota_B(\varphi)= \iota_B(\varphi^*)$ and $I_B(E,F^*)=(E\otimes_\lambda F)^*$ have no counterparts for completely integral mappings.