Ilijas Farah (York University)
TITRE / TITLE
Corona Rigidity
R脡SUM脡 / ABSTRACT
This story started with Weyl鈥檚 work on compact perturbations of pseudo-differential operators. The Weyl-von Neumann theorem asserts that two self-adjoint operators on a complex Hilbert space are unitarily equivalent modulo compact perturbations if and only if their essential spectra coincide. This was extended to normal operators by Berg and Sikonia. New impetus was given in the work of Brown, Douglas, and Fillmore, who replaced single operators with (separable) C*-algebras and compact perturbations with extensions by the ideal of compact operators. After passing to the quotient (the Calkin algebra, Q) and identifying an extension with a *-homomorhism into Q, analytic methods have to be supplemented with methods from algebraic topology, homological algebra, and (most recently) logic. Some attention will be given to the (still half-open) question of Brown-Douglas-Fillmore, whether Q has an automorphism that flips the Fredholm index. It is related to a very general question about isomorphisms of quotients, asking under what additional assumptions such isomorphism can be lifted to a morphism between the underlying structures. As general as it is, many natural instances of this question have surprisingly precise (and surprising) answers. This talk will be partially based on the preprint Farah, I., Ghasemi, S., Vaccaro, A., and Vignati, A. (2022). Corona rigidity. arXiv preprint arXiv:2201.11618 and some more recent results.
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