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
>  $U(H)$?
> 
> 
> 




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**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 $\ell^2(\mathbb N)$.



topologies on U(H)
The weak and ultraweak topologies coincide on bounded subsets of $B(H)$: see section 3.5 in Pedersen's book "C\*-algebras and their automorphism groups".


