Uniqueness of norm-preserving extensions of functionals on the space of compact operators - Martsinkevitš, Julia; Põldvere, Märt (MATHEMATICA SCANDINAVICA, 2019)
Godefroy, Kalton, and Saphar called a closed subspace $Y$ of a Banach space $Z$ an ideal if its annihilator $Y^\bot $ is the kernel of a norm-one projection $P$ on the dual space $Z^* $. If $Y$ is an ideal in $Z$ with respect to a projection on $Z^* $ whose range is norming for $Z$, then $Y$ is said to be a strict ideal. We study uniqueness of norm-preserving extensions of functionals on the space $\mathcal{K}(X,Y) $ of compact operators between Banach spaces $X$ and $Y$ to the larger space $\mathcal{K}(X,Z) $ under the assumption that $Y$ is a strict ideal in $Z$. Our main results are: (1) if $y^* $ is an extreme point of $B_{Y^{*} }$ having a unique norm-preserving extension to $Z$, and $x^{**} \in B_{X^{**} }$, then the only norm-preserving extension of the functional $x^{**} \otimes y^* \in \mathcal {K}(X,Y)^* $ to $\mathcal {K}(X,Z)$ is $x^{**} \otimes z^* $ where $z^* \in Z^* $ is the only norm-preserving extension of $y^* $ to $Z$; (2) if $\mathcal{K}(X,Y) $ is an ideal in $\mathcal{K}(X,Z) $ and $Y$ has Phelps' property $U$ in its bidual $Y^{**} $ (i.e., every bounded linear functional on $Y$ admits a unique norm-preserving extension to $Y^{**} $), then $\mathcal{K}(X,Y) $ has property $U$ in $\mathcal{K}(X,Z) $ whenever $X^{**} $ has the Radon-Nikodým property.