Abstract
Gaps in coronas are studied in this chapter. We show that the rich and well-studied gap spectrum of \(\mathscr P({\mathbb N})/ \operatorname {\mathrm {Fin}}\) 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 \(\bigoplus _{\aleph _1} {\mathbb Z}/2{\mathbb Z}\) 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.
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Farah, I. (2019). Gaps and Incompactness. In: Combinatorial Set Theory of C*-algebras. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-27093-3_14
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DOI: https://doi.org/10.1007/978-3-030-27093-3_14
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