There are many topologies on the algebra $B(H)$ of bounded operators on Hilbert space:
the weak, strong, ultraweak (also called σ-weak), ultrastrong (also called σ-strong), and some more...
Luckily, the weak and strong topologies agree when restricted to $U(H)\subset B(H)$.
Similarly, the ultraweak and ultrastrong topologies agree on $U(H)$.
Is it true that the weak and ultraweak topologies agree when restricted to $U(H)$?
A generalized sequence $a_i$ is weakly, strongly, ultraweakly, ultrastrongly convergent if:
• $\langle a_i\xi,\eta\rangle\to\langle a\xi,\eta\rangle\qquad \forall \xi,\eta\in H$
• $a_i\xi\to a\xi\qquad \forall \xi\in H$
• $\langle (a_i\otimes 1)\xi,\eta\rangle\to\langle (a\otimes 1)\xi,\eta\rangle\qquad \forall \xi,\eta\in H\otimes \ell^2(\mathbb N)$
• $(a_i\otimes 1)\xi\to (a\otimes 1)\xi\qquad \forall \xi\in H\otimes \ell^2(\mathbb N)$,
Here, $H\otimes \ell^2(\mathbb N)$ denotes the Hilbert space tensor product of $H$ and $\ell^2(\mathbb N)$.