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Let $K$ be a bounded closed convex subset of a Banach space $E$. A point $x$ in $K$ is called a diametral point if $$ \sup_{y\in K} \|x-y\|={\rm diam}(K). $$ where ${\rm diam}(K)$ denotes the diameter of $K$. The set $K$ is said to have normal structure if every non-trivial (i.e. containing at least two points) convex subset $H$ of $K$ contains a non-diametral point of $H$.

A dual Banach space $E$ has weak${}^*$ normal structure if every non-trivial weak* compact convex subset of $E$ has normal structure.

Does the dual space $B(\ell^\infty)$ of bounded linear operators on $\ell^\infty$ have weak* normal structure?

Recall that the natural predual of $B(\ell^\infty)$ is the space $\ell^\infty\hat{\otimes}\ell^1$ where $\hat{\otimes}$ denotes the projective tensor product.

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  • $\begingroup$ Could you give an example of a dual Banach space which does not have weakstar normal structure? $\endgroup$
    – Yemon Choi
    Commented Nov 13, 2010 at 18:18
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    $\begingroup$ The set $\lbrace (x_n): \sum_n x_n=1, x_n\ge 0\rbrace$ in $\ell^1$, as a dual of $c$, is such an example. $\endgroup$
    – TCL
    Commented Nov 21, 2010 at 12:57
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    $\begingroup$ Just observe that an affirmative answer would imply that every separable reflexive space has normal structure. $\endgroup$
    – Bill Johnson
    Commented Oct 1, 2014 at 19:15

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