## Annals of Functional Analysis

### 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 $Y^{\ast\ast}$ 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 $P$.

#### Article information

Source
Ann. Funct. Anal., Volume 10, Number 1 (2019), 46-59.

Dates
Accepted: 24 February 2018
First available in Project Euclid: 16 January 2019

https://projecteuclid.org/euclid.afa/1547629222

Digital Object Identifier
doi:10.1215/20088752-2018-0007

Mathematical Reviews number (MathSciNet)
MR3899955

Zentralblatt MATH identifier
07045484

#### Citation

Martsinkevitš, Julia; Põldvere, Märt. On the structure of the dual unit ball of strict $u$ -ideals. Ann. Funct. Anal. 10 (2019), no. 1, 46--59. doi:10.1215/20088752-2018-0007. https://projecteuclid.org/euclid.afa/1547629222

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