Banach spaces with weak*-sequential dual ball

Martínez-Cervantes, Gonzalo

doi:10.1090/proc/13843

Banach spaces with weak*-sequential dual ball
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by Gonzalo Martínez-Cervantes PDF

Proc. Amer. Math. Soc. 146 (2018), 1825-1832 Request permission

Abstract:

A topological space is said to be sequential if every subspace closed under taking limits of convergent sequences is closed. We consider Banach spaces with weak*-sequential dual ball. In particular, we show that if $X$ is a Banach space with weak*-sequentially compact dual ball and $Y \subset X$ is a subspace such that $Y$ and $X/Y$ have weak*-sequential dual ball, then $X$ has weak*-sequential dual ball. As an application we obtain that the Johnson-Lindenstrauss space $JL_2$ and $C(K)$ for $K$ a scattered compact space of countable height are examples of Banach spaces with weak*-sequential dual ball. These results provide a negative solution to a question of A. Plichko, who asked whether the dual ball of a Banach space is weak*-angelic whenever it is weak*-sequential.

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