All Automorphisms of the Calkin Algebra Are Inner

We prove that it is relatively consistent with the usual axioms of mathematics that all automorphisms of the Calkin algebra are inner. Together with a 2006 Phillips–Weaver construction of an outer automorphism using the Continuum Hypothesis, this gives a complete solution to a 1977 problem of Brown–Douglas–Fillmore. We also give a simpler and self-contained proof of the Phillips–Weaver result. Fix a separable infinite-dimensional complex Hilbert space H. Let B(H) be its algebra of bounded linear operators, K(H) its ideal of compact operators and C(H) = B(H)/K(H) the Calkin algebra. Let π : B(H) → C(H) be the quotient map. In [6, 1.6(ii)] (also [34], [42]) it was asked whether all automorphisms of the Calkin algebra are inner. Phillips and Weaver ([32]) gave a partial answer by constructing an outer automorphism using the Continuum Hypothesis. We complement their answer by showing that a well-known set-theoretic axiom implies all automorphisms are inner. Neither the statement of this axiom nor the proof of Theorem 1 involve set-theoretic considerations beyond the standard functional analyst’s toolbox. Theorem 1. Todorcevic’s Axiom, TA, implies that all automorphisms of the Calkin algebra of a separable Hilbert space are inner. Todorcevic’s Axiom (also known as the Open Coloring Axiom, OCA) is stated in §2.3. Every model of ZFC has a forcing extension in which TA holds ([40]). TA also holds in Woodin’s canonical model for negation of the Continuum Hypothesis ([43], [27]) and it follows from the Proper Forcing Axiom, PFA ([39]). The latter is a strengthening of the Baire Category Theorem and besides its applications to the theory of liftings it can be used to find other combinatorial reductions ([39, §8], [30]). The Calkin algebra provides both a natural context and a powerful tool for studying compact perturbations of operators on a Hilbert space. The original motivation for the problem solved in Theorem 1 comes from a classification problem for normal operators. By results of Weyl, von Neumann, Date: September 29, 2009. 1991 Mathematics Subject Classification. 46L40, 46L05, 03E75, 03E65 .