Kuiper's theorem via approximation

Question

Kuiper's theorem says that the unitary group $U(H)$ of a separable infinite dimensional Hilbert space $H$ is contractible, if it is equipped with the norm topology.

Let's suppose, I do not know this theorem, but I do know that $U^{st*}(H)$ is contractible, where the latter group is $U(H)$ equipped with the strong$^*$ topology generated by the semi-norms $p_v(a) = \lVert av \rVert$ and $q_v(a) = \lVert a^*v\rVert$ for all $v \in H$. I have a continuous map $U(H) \to U^{st*}(H)$.

Is there any way to see that this is a weak equivalence by an approximation argument, thereby proving Kuiper's theorem?

side remark:

The motivation for this question comes from a setup which looks completely different, but is from a certain point of view surprisingly similar: Let $\mathcal{O}_{\infty}$ be the Cuntz algebra on countably infinite generators. $Aut(\mathcal{O}_{\infty})$ carries two topologies: One from the norm via the inclusion into bounded maps on Banach spaces $\mathcal{O}_{\infty} \to \mathcal{O}_{\infty}$, the other is the so-called point-norm topology generated by the semi-norms $p_a(\alpha) = \lVert \alpha(a) \rVert$. It is known that $Aut(\mathcal{O}_{\infty})$ is weakly contractible in the latter topology, but I would like to know it for the first.

Answer 1

This is not an answer but too long for a comment.

It was shown in

Popa, S. and Takesaki,M., The Topological Structure of the Unitary and Automorphism Groups of a Factor, Commun. Math. Phys. 155, 93-101 (1993)

that the unitary group $U(R)$ of the hyperfinite $II_1$-factor $R$ is contractible in the strong topology (which is equal to the strong-$*$-topology in this case).

Years before, it was shown in

Araki, H., Smith, M.-S.B. and Smith, L., On the homotopical significance of the type of von Neumann algebra factors, Commun. Math. Phys. 22, 71-88 (1971)

that the first homotopy group of $U(R)$ in the norm toplogy is isomorphic to $(\mathbb R,+)$. Under this isomorphism, the element $\lambda \in [0,1]$ is represented by the loop $$[0,1] \ni t \mapsto \exp(2\pi i\cdot t p) \in U(R),$$ where $p\in R$ is some projection of trace $\lambda$.

This shows that the homotopy type depends heavily on the chosen topology and there is no reason (at least in general) that one should find an easy approximation argument.