Does the Calkin algebra admit an automorphism that induces the flip on ?
Background/Motivation: Let be a separable, infinite-dimensional Hilbert space. The Calkin algebra is the quotient of the bounded, linear operators on by the closed, two-sided ideal of compact operators. The important problem of whether the Calkin algebra has outer automorphisms was eventually shown to be independent of the usual set theoretic axioms ZFC: Phillips and Weaver, , proved that the Continuum Hypothesis implies that has many outer automorphism. This was complemented by a result of Farah, , who showed that the Open Coloring Axiom implies that all automorphisms of are inner.
Given a C*-algebra , every automorphism induces a group automorphism . It is known that . Hence, an automorphism of either induces the identity or the flip on . Every inner automorphism of a C*-algebra acts trivially on K-theory. Therefore, the result of Farah shows that assuming the Open Coloring Axiom, every automorphism of induces the identity on . As it turns out, the outer automorphisms constructed by Phillips and Weaver are locally inner, that is, for every element there exists a unitary such that . (The unitary depends on the element.) Since the generator of is represented by a unitary in , it follows that every locally inner automorphism of induces the identity on as well. One may consider automorphisms of that induce the flip on as ‚very outer‘, and the question is if such automorphisms exists (assuming some set-theoretic axioms).
- N.C. Phillips, N. Weaver, The Calkin algebra has outer automorphisms, Duke Math. J. 139 (2007) 185–202. https://doi.org/10.1215/s0012-7094-07-13915-2.
- I. Farah, All automorphisms of the Calkin algebra are inner, Ann. Math. 173 (2011) 619–661. https://doi.org/10.4007/annals.2011.173.2.1.