The Calkin algebra is the quotient of the C*-algebra of bounded linear operators on an infinite-dimensional separable complex Hilbert space by the closed ideal of compact operators. It is widely used in index theory and operator algebras.
It is sometimes viewed as a noncommutative analogue of the space of continuous functions on (where means taking the Stone-Čech compactification).
Todorcevic’s Axiom? (TA, also called the Open Coloring Axiom) implies that all automorphisms of the Calkin algebra are inner (Farah 2011a), whereas assuming the continuum hypothesis, one can construct an outer automorphism (Phillips-Weaver 2007). Note that TA contradicts CH (Rinot 2013), so these two options are mutually exclusive.
There are also some analogues of the above results where the separable Hilbert space is replaced by an arbitrary infinite-dimensional complex Hilbert space. For instance, the Proper Forcing Axiom? (PFA) implies that for every Calkin algebra, all automorphisms are inner (Farah 2011b).
Another result (Farah and Hirshberg 2016) of Farah and Hirshberg states that the Calkin algebra of a separable Hilbert space is not countably homogeneous?; in particular, it cannot be realized as a Fraïssé limit of metric structures.
N. Christopher Phillips, Nik Weaver, The Calkin algebra has outer automorphisms, Duke Math. J. Volume 139, Number 1 (2007), 185-202. arXiv:math/0606594
Ilijas Farah, All automorphisms of the Calkin algebra are inner, Annals of Mathematics 173 (2011), 619-661, doi: 10.4007/annals.2011.173.2.1 arXiv:0705.3085
Ilijas Farah, All automorphisms of all Calkin algebras, Math. Res. Lett. 18 (2011), no. 3, 489-503 arXiv:1007.4034
Ilijas Farah and Ilan Hirshberg, The Calkin algebra is not countably homogeneous, Proc. AMS, to appear. arXiv:1506.07455
Assaf Rinot, Open coloring and the cardinal invariant , (2013) blog post
Nik Weaver, Set theory and -algebras, The Bulletin of Symbolic Logic Vol. 13, No. 1 (Mar., 2007), pp. 1-20, arxiv:math/060198.LO