Gaps and incompactness

Abstract

Gaps in coronas are studied in this chapter. We show that the rich and well-studied gap spectrum of P(N) ∕ Fin embeds into the corona of every σ-unital, non-unital, C*-algebra. This is used to prove two incompactness results. The Choi–Christensen construction of Kadison–Kastler near, but nonisomorphic, C*-algebras is recast in terms of gaps: every gap in the Calkin algebra can be used to produce a family of examples of this sort. Every uniformly bounded representation of a countable, amenable, group in the Calkin algebra is unitarizable. Using a Luzin family, one defines a uniformly bounded, non-unitarizable, representation of (formula presented) in the Calkin algebra. This example yields an amenable operator algebra not isomorphic to a C*-algebra. This is a result of the author, Choi, and Ozawa.