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I added some results and references at Calkin algebra after I noticed that Zoran had added some comments about set-theoretic axioms leading to different properties. In particular the outer automorphism algebra of the Calkin algebra is trivial or not, depending on whether one has CH, or something that violates CH, Todocevic’s Axiom.
If I remember correctly the Calkin algebra is closely linked to the problem of Von Neumann on operators which were normal modulo compact ones, and which was solved by Brown, Douglas and Fillmore see Bron-Douglas-Fillmore theory. This linked to Steenrod K-homology and to strong shape theory, again if I remember correctly.
I cross-linked Brown–Douglas–Fillmore theory and Calkin algebra, and neatened up the former a little, adding ’related entries’ links.
Thanks Todd for fixing that! and David for linking. I wish I could see how this ties in with the CH stuff. There were indications at one time that Steenrod homotopy might link up with the old work of Osofsky on vanishing of derived functors of Lim but I never followed that up. It may not work that way.
It’s worth considering the analogue for the C*-algebra of the Stone-Cech remainder $\beta(\mathbb{N})\setminus \mathbb{N}$, which is apparently the question of whether non-trivial self-homeomorphisms exist; this is also independent of ZFC. CH is again a culprit in constructions: under CH there are only few trivial self-homeomorphisms, while at the same time allowing a strong structure theorem on the algebra, giving the needed cardinal inequality.
Given that $\beta(\mathbb{N})$ is an object that already has set theory axioms in play, it’s no wonder that axioms also have an effect here.
Weaver’s book Forcing for mathematicians is very nice in explaining this and many other examples in a concise way, that gave me a vague intuition why set theory makes itself known in what seems like a not-so-close area of study (I mean, this is standard operator algebras, not descriptive set theory, right?)
I’ve added a remark + reference to another result of Farah (and Hirshberg) that the Calkin algebra is not countably homogeneous (which I will fill in soon, since this now appears as a grey link in several pages): it fails to resemble a Fraïssé limit in the sense that an isomorphism of finitely-generated substructures does not generally extend to an automorphism of the entire thing.
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