Renorming $\mathcal{B}(\mathcal{H})$?

Consider the Banach space of all bounded operators $\mathcal{B}(\mathcal{H})$ on a (separable if you wish) Hilbert space $\mathcal{H}$ with the operator norm. Can we renorm this space to a strictly convex one? Recall that a Banach space is strictly convex whenever the unit sphere is the set of extreme points of the closed unit ball.

Seemingly, it is well known but I don't know how to prove that. I am interested in the cases $\mathcal{B}(E)$ ($E$ - some infinite dimensional Banach space; excluding the Argyros-Haydon space $E$ - in that case $\mathcal{B}(E)$ is separable, thus admits such a renorming) or $\mathcal{M}$ - a von Neumann algebra as well.

Solution 1:

In 1955 M.M. Day proved:

Theorem (Day) If $E$ is a separable Banach space then $E^{\ast}$ carries an equivalent strictly convex dual norm.

See Theorem 4 in M.M. Day, Strict convexity and smoothness of normed spaces, Trans. Amer. Math. Soc. 78 (1955), 516–528.

If $H$ is separable, the space $B(H)$ is the dual space of the separable space of trace class operators (the operators of finite rank are dense in the trace class operators), hence Day's theorem provides us with a positive answer to your question in the separable case, and for separable von Neumann algebras (i.e., those with separable pre-dual) as well.

The question on von Neumann algebras has a negative answer in general (loc. cit., Corollary to Theorem 8):

Theorem (Day, Phillips) If $I$ is an uncountable set then $\ell^{\infty}(I)$ does not admit an equivalent strictly convex norm.

By the results in Day's paper this also implies that $B(\ell^2(I))$ can't be renormed to a strictly convex space for $I$ uncountable: by item (3) at the bottom of page 518 the property of having a strictly convex renorming passes to closed subspaces; as $\ell^{\infty}(I)$ embeds into $B(\ell^2(I))$ this tells us that only for separable Hilbert spaces $H$ your question on $B(H)$ has a positive answer.

I do not know and did not really think about what one can say about $B(E)$, but let me mention the recent preprint

José Orihuela, Richard J. Smith, Stanimir Troyanski, Strictly convex norms and topology, arXiv:1012.5595v1

where you'll find an extensive discussion of related results and many pointers to the literature.