Question: Does the unitary group $U(\ell^2 \mathbb N)$, equipped with the strong operator topology, contain a countable dense subgroup which is amenable as a discrete group?
I would be also interested in answers to this for similar groups, such as for example for the Fredholm unitary group $U_C(\ell^2 \mathbb N)$ (with the uniform topology) or ${\rm Aut}(\mathbb Q)$ (with the topology of pointwise convergence). For ${\rm Aut}(\mathbb Q)$, this may be related (or even equivalent) to the question if Thompson's group $F$ is amenable; hence, this is probably a hard problem. For $U(\ell^2 \mathbb N)$, I do not even see a candidate for an amenable subgroup.
It is known that the set of pairs in $U(\ell^2 \mathbb N)$ which generate a finite subgroup is SOT-dense, so that there is no "local" obstruction to the generation of an amenable group; unlike for $U(\ell^2 \mathbb N)$ in the norm topology. Indeed, a pair of elements which is sufficiently norm-close to the generators of the left-regular representation of a non-abelian free group cannot generate an amenable group.
For $U(n)$, every amenable subgroup must be virtually solvable, so that it cannot be dense.
