On the structure of the dual unit ball of strict \(u\)-ideals (Q1738210)

Let $Y$ be a (closed) subspace of a Banach space $X$. If there is a norm-one projection $P : X^* \to X^*$ with $\ker P = Y^\perp$, then $Y$ is said to be an ideal in $X$. It is a strict ideal if $\text{ran} P$ is norming for $X$ and a $u$-ideal if $P$ satisfies $\|I - 2P\| = 1$. From \textit{G. Godefroy} et al. [Stud. Math. 104, No. 1, 13--59 (1993; Zbl 0814.46012)], we know that an ideal projection $P$ generates an operator $T_P : X \to Y^{**}$ such that $T_P(y) = y$ for all $y \in Y$ ($T_P$ is defined by $\langle T_p(x), y^* \rangle = \langle P(y^*), x \rangle$). The operator $T_P$ generates a topology $\tau_P = \sigma(Y^*,T_P(X))$ on $Y^*$, and $T_P$ is an isometry if and only if the projection $P$ is associated with a strict ideal. \par It is known that, if $X$ is a $u$-ideal in $X^{**}$, then $X$ is a strict $u$-ideal in $X^{**}$ if and only if $B_{X^*}$ is the norm-closed convex hull of its weak$^*$-denting points. Note that weak$^*$-denting points of $B_{X^*}$ have unique norm-preserving extension to $X^{**}$. \par A natural question is if something similar can be said if $Y$ is a strict $u$-ideal in $X$. \par It is shown that, if $Y$ is a strict $u$-ideal in a Banach space $X$ with respect to an ideal projection $P$ on $X^*$, and $X/Y$ is separable, then $B_{Y^*}$ is the $\tau_P$-closed convex hull of functionals admitting a unique norm-preserving extension to $X$. The proofs consists of a detailed study of slices and generalizations of denting points. These results are also used to give a new proof of the fact that, if $Y$ is a strict $u$-ideal in $X$, then every ideal projection for $Y$ in $X$ is strict.