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 $U(H)$?

**Definitions:**

A generalized sequence $a_i$ is

*weak*ly,

*strong*ly,

*ultraweak*ly,

*ultrastrong*ly convergent if:

• $\langle a_i\xi,\eta\rangle\to\langle a\xi,\eta\rangle\qquad \forall \xi,\eta\in H$

• $a_i\xi\to a\xi\qquad \forall \xi\in H$

• $\langle (a_i\otimes 1)\xi,\eta\rangle\to\langle (a\otimes 1)\xi,\eta\rangle\qquad \forall \xi,\eta\in H\otimes \ell^2(\mathbb N)$

• $(a_i\otimes 1)\xi\to (a\otimes 1)\xi\qquad \forall \xi\in H\otimes \ell^2(\mathbb N)$,

respectively.

Here, $H\otimes \ell^2(\mathbb N)$ denotes the Hilbert space tensor product of $H$ and $\ell^2(\mathbb N)$.