It is sometimes viewed as a noncommutative analogue of the space of continuous functions on $\beta(\omega) \,\backslash\, \omega$ (where $\beta$ means taking the Stone-Čech compactification).

Properties

General

Some results invoking the Calkin algebra rely on extra axioms in set theory, see for instance (Weaver 2007) for a sample of such results.

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 $\mathfrak{b}$, (2013) blog post

Nik Weaver, Set theory and $C^\ast$-algebras, The Bulletin of Symbolic Logic Vol. 13, No. 1 (Mar., 2007), pp. 1-20, arxiv:math/060198.LO