On norm bounded sets of $B(H)$, the weak (operator) and ultraweak topologies coincide.

Question

I am a bit confused of the naming of all these topologies. In our script we defined the $\sigma$-weak topology as the topology determined by the seminorms: $$P_{{(x_n,y_n)}_{n\in \mathbb{N}}}(A) = \sum_n |<x_n,Ay_n>| $$ where the two sequences $x_n$ and $y_n$ have finite 2-norm. As far I can tell the ultraweak and $\sigma$weak topology in literature are the same. I want to prove the statement:
On norm bounded sets of $B(H)$, the weak (operator) and ultraweak topologies coincide. but i have no clue where to begin. Thanks for any help. Btw I already looked into Banach Alaoglu but its not clear how to use it.

Answer 1

This problem doesn't require heavy machinery, just a couple of nets. Indeed, it suffices to show the following:

Let $A\in B(H)$ be given, and let $(A_\lambda)_{\lambda\in\Lambda}$ be a net in $B(H)$ for which there is some $M>0$ such that $\|A_\lambda\|\le M$ for all $\lambda\in\Lambda$. Then $(A_\lambda)_{\lambda\in\Lambda}$ converges to $A$ in the weak operator topology if and only if it converges to $A$ in the ultraweak topology.

Having properly formulated the problem, it is not difficult to see that ultraweak convergence implies convergence in the weak operator topology (no need for the boundedness condition here). For the converse direction, take two sequences $(x_n)_{n\in\mathbb{N}}$ and $(y_n)_{n\in\mathbb{N}}$ having finite $2$-norm, use the boundedness condition to truncate your sequences, and use weak operator convergence on each term in the truncated sequences to show that $P_{(x_n,y_n)_{n\in\mathbb{N}}}(A_\lambda)\to P_{(x_n,y_n)_{n\in\mathbb{N}}}(A)$.