Approximation properties for dual spaces

Abstract

We prove that a Banach space $X$ has the metric approximation property if and only if $\mathcal F(Y,X)$ is an ideal in $\mathcal L(Y,X^{**})$ for all Banach spaces $Y$. Furthermore, $X^*$ has the metric approximation property if and only if for all Banach spaces $Y$ and all Hahn-Banach extension operators $\phi : X^* \rightarrow X^{***}$ there exists a Hahn-Banach extension operator $\Phi : {\mathcal F(Y,X)}^* \rightarrow {\mathcal L(Y,X^{**})}^*$ such that $\Phi(x^* \otimes y^{**}) = (\phi x^*) \otimes y^{**}$ for all $x^* \in X^*$ and all $y^{**} \in Y^{**}$. We also prove that $X^*$ has the approximation property if and only if for all Banach spaces $Y$ and all Hahn-Banach extension operators $\phi : X^* \rightarrow X^{***}$ there exists a Hahn-Banach extension operator $\Phi : {\mathcal F(Y,X)}^* \rightarrow {\mathcal W(Y,X^{**})}^*$ such that $\Phi(x^* \otimes y^{**}) = (\phi x^*) \otimes y^{**}$ for all $x^* \in X^*$ and all $y^{**} \in Y^{**}$, which in turn is equivalent to $\mathcal F(Y,\hat{X})$ being an ideal in $\mathcal W(Y,\hat{X}^{**})$ for all Banach spaces $Y$ and all equivalent renormings $\hat{X}$ of $X$