Julia Martsinkevitš, Märt Põldvere. Source: Annals of Functional Analysis, Volume 10, Number 1, 46--59.Abstract:
It is known that if a Banach space $Y$ is a $u$ -ideal in its bidual $Y^{\ast\ast}$ with respect to the canonical projection on the third dual $Y^{\ast\ast\ast}$ , then $Y^{\ast}$ contains “many” functionals admitting a unique norm-preserving extension to $Y^{\ast\ast}$ —the dual unit ball $B_{Y^{\ast}}$ is the norm-closed convex hull of its weak $^{\ast}$ 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 $X^{\ast}$ , and $X/Y$ is separable, then $B_{Y^{\ast}}$ 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 $Y^{\ast}$ defined by the ideal projection $P$ .