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, [1], proved that the Continuum Hypothesis implies that
has many outer automorphism. This was complemented by a result of Farah, [2], 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).
- [1]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.
- [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.