3
$\begingroup$

You could reference Dixmier's book on Von Neumann Algebras p.42 Theorem 1 and its proof to know the entirety of the context. Otherwise, the most relevant things are below:

  1. Let $M$ be an ultraweakly closed subspace of $B(H)$ and $K$ a convex subset of $M$. Then he claims $K$ is ultraweakly closed if and only if $K\cap M_r$ is ultraweakly closed for all balls $M_r$ of radius $r$ in $M$ centered at $0$. Former implies latter is clear. What about the other way? He doesn't really make an attempt to prove this and instead references some outside fact and a seemingly unrelated subclaim. I would not mind if an entirely new proof were suggested in the answers, although a filling in of the missing details would be equally appreciated.

  2. He then uses this fact to prove that if $\phi$ is a linear form on $M$, continuous in norm, for which its restriction to the unit ball is continuous in ultrastrong topology, then $\phi$ is actually ultraweakly continuous. Along the way, he concludes (and I agree with this given 1.) that $\phi^{-1}({0})$ is ultraweakly closed. I just don't see why this implies the conclusion, even though he asserts it without further comment in the next step. Thanks!

  • 0
    Linear algebra tells us that $M/\phi^{-1}(0)$ is one-dimensional if $\phi \neq 0$. There's only one Hausdorff vector space topology on a finite-dimensional space and with respect to that topology all linear maps from the space are continuous.2012-07-16

0 Answers 0