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, and . This chapter ends with a convenient structure result for comeagre subsets of products of finite spaces.
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- 1.
This phenomenon can be viewed from the operator-algebraic point of view, see [47, §II].
- 2.
Hence a more accurate, but not particularly catchy, name for this property would be “the nonempty intersection of all finite subfamilies property”.
- 3.
In the literature χA is sometimes denoted 1A.
- 4.
- 5.
In the sense that every infinite \(\mathsf {X}\subseteq {\mathbb N}\) has an infinite subset in the ideal.
- 6.
At this point, it may not be clear what gaps have to do with anything except a curiosity (Exercise 9.10.6) and a failed attempt to prove the Continuum Hypothesis. For now, we will leave this gun hanging on the wall, waiting for its moment to fire.
- 7.
This encounter was only implicit since we have not used the terminology. It was also a “non-encounter” since the η1 sets were defined by postulating the absence of small gaps, small limits, and endpoints or “jumps”.
- 8.
Following von Neumann’s convention that j = {0, …, j − 1}, one could write \(p^{ \operatorname {\mathrm {{\mathbf {K}}}}}_j\) for \(p^{ \operatorname {\mathrm {{\mathbf {K}}}}}_{[0,j)}\); however this notation is avoided as \(p^{ \operatorname {\mathrm {{\mathbf {K}}}}}_j\) could easily be confused with \(p^{ \operatorname {\mathrm {{\mathbf {K}}}}}_{\{j\}}\).
- 9.
The letter \(\mathfrak c\) has already been taken, for \(2^{\aleph _0}\).
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Farah, I. (2019). Set Theory and Quotients. In: Combinatorial Set Theory of C*-algebras. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-030-27093-3_9
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