Takesaki theorem 2.6

Question

Consider the following theorem in Takesaki's book "Theory of operator algebras". In particular, I'm focussed on understanding (ii). It is not mentioned in the picture, but $S$ is the unit ball of $\mathscr{L}(\mathfrak{h}).$

I can see that $(ii.1) \iff (ii.2)\iff (ii.3)\iff (ii.4)$ and that $(ii.5)\implies (ii.6)\implies (ii.7)$. Also, since the 'weaker' topologies agree with their $\sigma$-counterparts on bounded subsets, it is also clear that $(ii.x)\implies (ii.x+4)$ for $x=1,2,3$.

However, why do the other implications hold? The author just says that this follows by general theory of topological vector spaces, so if anyone can fill in the gap that would be great. In particular, I think it will be sufficient to know how I can prove $(ii.7)\implies (ii.3).$

Answer 1

This question has been asked afterwards in Math Overflow, where it received three very good answers.

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I don't know how Takesaki thinks of this. I do know that it takes Kadison-Ringrose several pages to prove the implication (ii.6)$\implies$(ii.4).

Roughly, they show

• that strongly continuous on the ball implies normal
• that normal (via complete additivity) implies (ii.4). This is Theorem 7.1.9 in Kadison-Ringrose II (and the result comparable to the one you quote is 7.1.12). The proof (including 7.1.8, that is used) is about 2 pages long.