I am trying to make an excercise given during lectures on operator algebras.
Let $A\subset\mathcal{L}(\mathcal{H})$ be a concrete (subset of operators on a Hilbert space) unital C*-algebra. Let $M=A''$ the bicommutant of $A$. Show that for every unitary $u$ in $M$, there exists a net of unitaries $v_\lambda\in A$ such that $v_\lambda\to u$ in the strong operator topology.
There is the extra hint that you should remember that $e^{it}=\cos(t)+i\sin(t)$ for $t\in \mathbb{R}$.
What I tried so far is the following.
First of all note that $M$ is a von Neumann algebra, as a consequence of the bicommutant theorem. For this we can show that every unitary is of the form $\exp(ih)$ for some $h\in M$ self-adjoint and $\lvert\lvert h\rvert\rvert\leq \pi$, using the Borel functional calculus. We also have by the Kaplinsky density theorem that the ball of radius $\pi$ of self-adjoints in $A$ lies dense inside the ball of radios $\pi$ of self-adjoints in $M$. We can therefore find a net $h_\lambda\in A$ of self-adjoints with radius at most $\pi$. Using continuous functional calculus we can define $\exp(ih_\lambda)$ and it is not hard to see these are unitaries.
To conclude I would need to show that for all $\xi\in\mathcal{H}$, we have $$\lvert\lvert(\exp(ih)-\exp(ih_\lambda))(\xi)\rvert\rvert\to 0$$ but I do not really know how I can say anything about this norm. Does anyone know how to proceed and if this is the right approach?
It is sufficient to show that $\lim_{t\to\pm\infty}\exp(it)/t=0$ in order for $t\mapsto exp(it)$ to be strongly continuous. This clearly holds.
Since $h_\lambda$ is a net of self-adjoints converging to $h$ in the operator topology, it converges to $h$ in the strong operator topology. Hence, the result follows by applying the definition of the $f$ being strongly continuous.