On the structure of the dual unit ball of strict u-ideals

Abstract

It is known that if a Banach space $Y$ is a $u$-ideal in its bidual $Y^{**}$ with respect to the canonical projection on the third dual $Y^{***}$ , then $Y^{*}$ contains “many” functionals admitting a unique norm-preserving extension to $Y^{**}$—the dual unit ball $B_{Y^{*}}$ is the norm-closed convex hull of its weak${}^{*}$ strongly exposed points by a result of Å. Lima from 1995. We show 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 $X$, where ${\tau }_P$ is a certain weak topology on $Y^{*}$ defined by the ideal projection $P$.