Continuity of $*$-map under different topology on $B(H)$

Question

Let $H$ be a Hilbert space and $\phi:B(H) \to B(H)$ is the involution map, that is $\phi(x)=x^*$ for all $x \in B(H).$ Now I know that $\phi$ is continuous with respect to weak (operator) topology and is not continuous with respect to Strong (operator) topology. Now my question: what about with respect to ultraweak topology? Basically, if $(u_i)_{i \in I}$ a net in $B(H)$ such that $u_i \to u$ under ultraweak topology in $B(H)$, is it possible to show that $u_i^* \to u^*$ under ultraweak topology in $B(H)$? Or, can you please help me with a counterexample for which $\phi$ is not continuous with respect to ultarweak topology on $B(H)$?
Thank you for your support.

Answer 1

The ultraweak topology is determined by the functionals of the form $A\mapsto {\rm Tr}(TA),$ where $T$ is a trace class operator. Also $T^*$ is of trace class. Let $A_\alpha\to A$ in ultraweak topology. Then $$ {\rm Tr}(TA_\alpha^*)={\rm Tr}(A_\alpha^*T)=\overline{{\rm Tr}(T^*A_\alpha)}\underset{\alpha}{\longrightarrow} \overline{{\rm Tr}(T^*A)}={\rm Tr}(A^*T)={\rm Tr}(TA^*)$$ Hence $A_\alpha^*\to A^*$ in the ultraweak topology.