Automorphisms of the Calkin algebra

Von | Juni 20, 2020

Does the Calkin algebra \mathcal{Q} admit an automorphism that induces the flip on  K_1(\mathcal{Q})?

Background/Motivation: Let H be a separable, infinite-dimensional Hilbert space. The Calkin algebra \mathcal{Q}:=\mathcal{B}(H)/\mathcal{K}(H) is the quotient of the bounded, linear operators on H 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 \mathcal{Q} 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 \mathcal{Q} are inner.

Given a C*-algebra A, every automorphism \alpha\colon A\to A induces a group automorphism K_1(A)\to K_1(A). It is known that K_1(\mathcal{Q})\cong\mathbb{Z}. Hence, an automorphism of \mathcal{Q} either induces the identity or the flip on K_1(\mathcal{Q}). 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 \mathcal{Q} induces the identity on K_1(\mathcal{Q}). As it turns out, the outer automorphisms \alpha constructed by Phillips and Weaver are locally inner, that is, for every element a\in\mathcal{Q} there exists a unitary u\in\mathcal{Q} such that \alpha(a)=uau^*. (The unitary depends on the element.) Since the generator of K_1(\mathcal{Q}) is represented by a unitary in \mathcal{Q}, it follows that every locally inner automorphism of \mathcal{Q} induces the identity on K_1(\mathcal{Q}) as well. One may consider automorphisms of \mathcal{Q} that induce the flip on K_1(\mathcal{Q}) as ‚very outer‘, and the question is if such automorphisms exists (assuming some set-theoretic axioms).

  1. [1]
    N.C. Phillips, N. Weaver, The Calkin algebra has outer automorphisms, Duke Math. J. 139 (2007) 185–202.
  2. [2]
    I. Farah, All automorphisms of the Calkin algebra are inner, Ann. Math. 173 (2011) 619–661.

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