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Mapping Spaces

Let $ E$, $ F$ be Banach spaces. We consider a linear subspace $ A(E,F)$ of the space $ B(E,F)$ of the continuous operators between $ E$ and $ F$ which contains all finite rank maps and is a Banach space with respect to a given norm. Furthermore, it is usually required that $ A(E,F)$ be defined for all pairs of Banach spaces $ E$ and $ F$. Such a space is called - according to A. Grothendieck - a mapping space .

Analogously, we call an operator space $ A(X,Y)$ which is a linear subspace of $ \mathit{CB}(X,Y)$ a $ \mathit{CB}$-mapping space. Note that generally the algebraic identification of $ M_n(A(X,Y))$ with $ A(X,M_n(Y))$ fails to be isometric and that the norms on $ A(X,M_n(Y))$ do not generate an operator space structure for $ A(X,Y)$.

There is a close relationship between mapping spaces and tensor products: The space $ F(X,Y)$ of all finite rank maps between $ X$ and $ Y$ and the algebraic tensor product of $ X^*$ with $ Y$ are isomorphic:

$\displaystyle X^*\otimes_{\mathrm{alg}} Y\cong F(X,Y).
$

This identification enables us to transfer norms from one space to the other one. To this end, we consider the extension of the mapping $ X^*\otimes Y\to F(X,Y)$ to the completion with respect to an operator space tensor norm $ X^*\widetilde{\otimes} Y$ :

$\displaystyle \Phi: X^*\widetilde{\otimes} Y \rightarrow \mathit{CB}(X,Y)$.$\displaystyle $

$ \Phi$ is in general neither injective nor surjective. As a CB-mapping space one obtains

$\displaystyle \mathrm{Im}(\Phi) \subset \mathit{CB}(X,Y)$

with the operator space norm of

$\displaystyle (X^*\tilde{\otimes} Y)/ \mathrm{Ker} (\Phi)$   .

We consider now assignments that assign a mapping space $ A(\cdot,\cdot)$ with operator space norm $ \alpha(\cdot)$ to every pair of operator spaces. In the Banach space theory A. Pietsch intensified the notion of mapping spaces to that of the operator ideals [Pie78]. Analogously, we consider operator ideals which are mapping spaces with the $ \mathit{CB}$-ideal property [ER94], i.e , the composition
$\displaystyle \mathit{CB}(X_1,X_2) \times A(X_2,Y_2) \times \mathit{CB}(Y_2,Y_1)$ $\displaystyle \rightarrow$ $\displaystyle A(X_1,Y_1)$  
$\displaystyle (\Psi_1,\Phi,\Psi_2)$ $\displaystyle \mapsto$ $\displaystyle \Psi_2 \circ \Phi \circ \Psi_1$  

is for all operator spaces $ X_1$, $ X_2$, $ Y_1$, $ Y_2$ defined and jointly completely contractive .

A $ \mathit{CB}$-ideal is called local [EJR98], if its norm satisfies:

$\displaystyle \alpha(\varphi)=\sup\{\alpha
(\varphi \vert _L):L\subset X ,\; \dim L<\infty\}$.$\displaystyle $



Subsections
next up previous contents index
Next: Completely nuclear mappings Up: What are operator spaces? Previous: -extreme points   Contents   Index
Prof. Gerd Wittstock 2001-01-07