Let $N$ be a II$_1$-factor. I want to prove that $N$ is simple as a C*-algebra, i.e. every non-trivial norm closed ideal $I$ in $N$ is already $N$.
This can be done be using Dixmiers approximation theorem, however I think that there should be some straightforward approach.
I'll be glad to be proven wrong, but I don't see why you would expect a "straightforward approach". Because the centre is trivial, any norm closed ideal $J$ in a II$_1$-factor $M$ will be sot-dense. So you need a tool that can show that a proper sot-dense, norm-closed ideal cannot exist.
Dixmier's Approximation Theorem is a miracle here that allows you to show that the norm-closed unitary orbit of any element contains an element of the centre. Any other technique that shows that $M$ is simple would allow you to prove that the norm closure of $MxM$ contains a scalar, for any $x\in M$. It doesn't look trivial to me.
Edit: Mateusz Wasilewski has proven me wrong, by noticing that this is Theorem 6.8.3 in Kadison-Ringrose (in the second volume). The proof goes along the line of the proof that the compacts are the only ideal of $B(H)$ (although it is not that simple!).