Let $K$ be a bounded closed convex subset of a Banach space $E$. A point $x$ in $K$ is called a diametral point if $$ \sup_{y\in K} ||x-y||=diam(K). $$ where $diam(K)$ denotes the diameter of $K$. The set $K$ is said to have normal structure if every nontrivial (i.e contains 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 nontrivial weak* compact convex subset of $E$ has normal structure.

Does the dual space $B(\ell^\infty)$ of bounded linear operators on $\ell^\infty$ has 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.

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.