Chris Phillips and Nik Weaver wrote a paper called *The Calkin Algebra has Outer Automorphisms*, where they showed that the Continuum Hypothesis implies that the Calkin algebra $\mathcal{B(H)/K(H)}$ has outer automorphisms.  See also [this paper][1] of Farah, McKenney, and Schimmerling, and references therein; they show that it is relatively consistent with ZFC that the Calkin algebra has only inner automorphisms, and hence the question of existence of outer automorphisms is independent of ZFC.


  [1]: https://arxiv.org/abs/1211.4134 "Some Calkin algebras have outer automorphisms"