Example of a wot convergent net but not $\sigma -$ weak convergent

Question

Let $B(H)$ be the space of bounded linear operators. Define the $\sigma-$ weak topology on it by seminorms $p_{h,k} (x)=|\sum_{n\geq 1}(xh_n,k_n)$ where $h=\{h_n\}\subset H ,~~ k=\{k_n\}\subset H $ and $\sum \|h_n\|^2<\infty , ~~~\sum\|k_n\|^2<\infty$. Also Define weak operator topology on $B(H)$ by seminorms $p_{h,k}(x)=|(xh,k)|$ for $h,k\in H$. I know that $\sigma- $ weak topology is stronger that wot.For see that we have strict inclusion I need an example of a weak operator topology convergent net but not $\sigma-$ weak convergent. Please help me. Thank you.

Answer 1

Here is an example of a net that is weakly convergent, but not ultraweakly convergent.

I will use the fact that $X_j\to0$ ultraweakly precisely when $\text{Tr}(AX_j)\to0$ for all trace-class operators $A$.

Fix $A$ to be an injective trace-class operator, so that $AP\ne0$ for all projections $P$. For example, you could take $A=\sum_k\frac1{k^2}\,\langle\, \cdot\, e_k,e_k\rangle$ for a given orthonormal basis.

Let $\mathcal F=\{F\subset H:\ F \text{ is a finite-dimensional subspace} \}$, ordered by inclusion. We construct a net of operators indexed by $\mathcal F$ as follows: $$ T_F=\frac1{\text{Tr}(AP_{F^\perp})}\,P_{F^\perp}, $$ where $P_{F^\perp}$ is the orthogonal projection onto $F^\perp$. Then, for any $x\in H$, if we move far enough along the net we will have $x\in F$, so $T_Fx=0$, and then $T_F\to0$ in the wot topology (and the sot, too!).

But, for any $F$, $$ \text{Tr}(AT_F)=\frac{\text{Tr}(AP_{F^\perp})}{\text{Tr}(AP_{F^\perp})}=1. $$ So $\{T_F\}$ does not converge ultraweakly.