Does the dual Banach space $B(\ell^\infty)$ has weak* normal structure? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T21:11:17Z http://mathoverflow.net/feeds/question/45911 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/45911/does-the-dual-banach-space-b-ell-infty-has-weak-normal-structure Does the dual Banach space $B(\ell^\infty)$ has weak* normal structure? BigBill 2010-11-13T12:02:45Z 2010-11-13T12:02:45Z <p>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 $H$.</p> <p>A dual Banach space $E$ has weak* normal structure if every nontrivial weak* compact convex subset of $E$ has normal structure.</p> <blockquote> Does the dual space $B(\ell^\infty)$ of bounded linear operators on $\ell^\infty$ has weak* normal structure? </blockquote> <p>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.</p>