The classical topic of automorphisms of the Boolean algebra P(N)/Fin is one of the most exciting chapters of the set-theoretic research. One of its achievements is the result saying that assuming PFA or OCA+MA any such automorphism is trivial i.e., induced by a bijection between two cofinite subsets of N. It follows that the same is true for the commutative Banach algebra $\ell_\infty/c_0$ or C(N*). I. Farah recently transported these ideas to the non-commutative context proving that under OCA+MA all automorphisms of the C*-algebra B(H)/K(H) (bounded operators on separable Hilbert space divided by compact operators, i.e., the Calkin algebra) are inner (of the form a(x)=yxy* for a unitary y).
In my talk I will present some of the results of the research conducted last year with Cristobal Rodriguez-Porras from Universite Paris 7, a doctoral student of Boban Velickovic and myself concerning automorphisms of the Banach space $\ell_\infty/c_0$ or C(N*). Such automorphisms do not need to preserve the multiplication (or intersections) but they just need to be linear and bounded.
We obtain the dependence of some of the basic phenomena
concerning these automorphisms on combinatorial principles like CH or OCA+MA but the emerging picture is quite different than in the case of the other stuctures, some "chaos" already exists in ZFC and does not require CH and one may hope only for local consistency non-chaos results rather than global. The talk will have a survey character and I will not show details of any proofs. A 50-page preprint is in the final stage
of preparation.
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