Operator algebras and Logic seminar, I: Ilijas Farah

Speaker: Ilijas Farah (York University)

Title: Outer automorphisms of the Calkin algebra

Abstract: The Calkin algebra Q is the quotient of the algebra B(H) of all bounded linear operators on the infinite-dimensional separable complex Hilbert space H over the ideal of compact operators. The image s of the unilateral shift on an orthonormal basis of H in Q is a unitary operator.

In early 1970s, Brown, Douglas, and Fillmore asked whether Q has an automorphism that conjugates s to its adjoint s*. Such automorphism would have to be outer. While the BDF question is still open, it is known that the Continuum Hypothesis implies the existence of an outer automorphism of Q (Phillips--Weaver) and that forcing axioms imply that all automorphisms of Q are inner (F.). The latter result thus implies that, consistently with ZFC, the BDF question has a negative answer. I will present a self-contained construction of an outer automorphism of Q from the Continuum Hypothesis. Lamentably, all known outer automorphisms of Q have the property that their restriction to every separable C*-algebra is implemented by a unitary, and therefore cannot conjugate s to s*.