Reasonable non--Radon--Nikodym ideals

For any abelian Polish sigma-compact group H there exist a sigma-ideal Z over N and a Borel Z-approximate homomorphism f : H --> H^N which is not Z-approximable by a continuous true homomorphism g : H --> H^N

Logic and C∗\mathrm{C}^*-algebras: set theoretical dichotomies in the theory of continuous quotients

Given a nonunital $\mathrm{C}^*$-algebra $A$ one constructs its corona algebra $\mathcal M(A)/A$. This is the noncommutative analog of the \v{C}ech-Stone remainder of a topological space. We analyze the two faces of these algebras: the first one is given assuming CH, and the other one arises when Forcing Axioms are assumed. In their first face, corona $\mathrm{C}^*$-algebras have a large group of automorphisms that includes nondefinable ones. The second face is the Forcing Axiom one; here the automorphism group of a corona $\mathrm{C}^*$-algebra is as rigid as possible, including only definable elementsComment: This is the author's Ph.D. thesis, defended in April 2017 at York University, Toront

Functorial rings of quotients—III: the maximum in archimedean f-rings

AbstractThe category of discourse is Arf, consisting of archimedean f-rings with identity and ℓ-homomorphisms which preserve the identity. Based on a notion of Wickstead, an f-ring A is said to be strongly ω1-regular if for each countable subset D⊆A of pairwise disjoint elements there is an s∈A such that d2s=d, for each d∈D, and xs=0, for each x∈A which annihilates each d∈D. It is shown that strong ω1-regularity is monoreflective in Arf; indeed, A is strongly ω1-regular if and only if it is laterally σ-complete and has bounded inversion, if and only if A is von Neumann regular and laterally σ-complete. Recently the authors have characterized the category of laterally σ-complete archimedean ℓ-groups with weak unit as the epireflective class generated by the class of all laterally complete archimedean ℓ-groups. This, together with the above characterization of strong ω1-regularity, leads to a description of the subcategory upon which the maximal functorial ring of quotients μ(Q) in Arf reflects

The definable content of homological invariants I: Ext\mathrm{Ext} & lim1\mathrm{lim}^1

This is the first installment in a series of papers in which we illustrate how classical invariants of homological algebra and algebraic topology can be enriched with additional descriptive set-theoretic information. To effect this enrichment, we show that many of these invariants can be naturally regarded as functors to the category of groups with a Polish cover. The resulting definable invariants provide far stronger means of classification. In the present work we focus on the first derived functors of $\mathrm{Hom}(-,-)$ and $\mathrm{lim}(-)$. The resulting definable $\mathrm{Ext}(B,F)$ for pairs of countable abelian groups $B,F$ and definable $\mathrm{lim}^{1}(\boldsymbol{A})$ for towers $\boldsymbol{A}$ of Polish abelian groups substantially refine their classical counterparts. We show, for example, that the definable $\textrm{Ext}(-,\mathbb{Z})$ is a fully faithful contravariant functor from the category of finite rank torsion-free abelian groups $\Lambda$ with no free summands; this contrasts with the fact that there are uncountably many non-isomorphic such groups $\Lambda$ with isomorphic classical invariants $\textrm{Ext}(\Lambda,\mathbb{Z})$. To facilitate our analysis, we introduce a general Ulam stability framework for groups with a Polish cover and we prove several rigidity results for non-Archimedean abelian groups with a Polish cover. A special case of our main result answers a question of Kanovei and Reeken regarding quotients of the $p$-adic groups. Finally, using cocycle superrigidity methods for profinite actions of property (T) groups, we obtain a hierarchy of complexity degrees for the problem $\mathcal{R}(\mathrm{Aut}(\Lambda)\curvearrowright\mathrm{Ext}(\Lambda,\mathbb{Z}))$ of classifying all group extensions of $\Lambda$ by $\mathbb{Z}$ up to base-free isomorphism, when $\Lambda =\mathbb{Z}[1/p]^{d}$ for prime numbers $p$ and $d\geq 1$.Comment: Typos fixed, and a table of contents adde