Set Theory and Quotients

Abstract

Several themes picked up later on in this text originate in this chapter. The first one is the structure of the Boolean algebra \(\mathcal P({\mathbb N})/ \operatorname {\mathrm {Fin}}\) and related quotient structures. The interplay between separability of \(\mathcal P({\mathbb N})\) (identified with the Cantor space) and countable saturation of \(\mathcal P({\mathbb N})/ \operatorname {\mathrm {Fin}}\) is used to construct several objects witnessing the incompactness of ℵ1,such as the independent families, almost disjoint families, and gaps in \(\mathcal P({\mathbb N})/ \operatorname {\mathrm {Fin}}\). In the latter sections this Boolean algebra is injected into massive corona C∗-algebras.This is used to construct subalgebras of \(\mathcal B(H)\) with unexpected properties, such as an amenable norm-closed algebra of operators on a Hilbert space not isomorphic to a C∗-algebra (Section 15.5), and Kadison–Kastler near, but not isomorphic, C∗-algebras (Section 14.4). We introduce the Rudin–Keisler ordering on the ultrafilters and construct Rudin–Keisler incomparable nonprincipal ultrafilters on \({\mathbb N}\). Basics of the Tukey ordering of directed sets are presented in Section 9.6. We prove that two directed sets are cofinally equivalent if and only if they are isomorphic to cofinal subsets of some directed set. We study the directed set \({{\mathbb N}}^{{\mathbb N}}\), the associated small cardinals \({\mathfrak b}\) and \(\mathfrak d\), and two directed sets cofinally equivalent to \({{\mathbb N}}^{{\mathbb N}}\) used to stratify the Calkin algebra, Open image in new window and Open image in new window. This chapter ends with a convenient structure result for comeagre subsets of products of finite spaces.