On the length of chains of proper subgroups covering a topological group

We prove that if an ultrafilter L is not coherent to a Q-point, then each analytic non-sigma-bounded topological group G admits an increasing chain <G_a : a of its proper subgroups such that: (i) U_{a in b(L)} G_a=G; and $(ii)$ For every sigma-bounded subgroup H of G there exists a such that H is a subset of G_a. In case of the group Sym(w) of all permutations of w with the topology inherited from w^w this improves upon earlier results of S. Thomas

Trivial automorphisms

We prove that the statement `For all Borel ideals I and J on $\omega$, every isomorphism between Boolean algebras $P(\omega)/I$ and $P(\omega)/J$ has a continuous representation' is relatively consistent with ZFC. In this model every isomorphism between $P(\omega)/I$ and any other quotient $P(\omega)/J$ over a Borel ideal is trivial for a number of Borel ideals I on $\omega$. We can also assure that the dominating number is equal to $\aleph_1$ and that $2^{\aleph_1}>2^{\aleph_0}$. Therefore the Calkin algebra has outer automorphisms while all automorphisms of $P(\omega)/Fin$ are trivial. Proofs rely on delicate analysis of names for reals in a countable support iteration of suslin proper forcings.Comment: Thoroughly revised versio

Cardinal invariants related to permutation groups

AbstractWe consider the possible cardinalities of the following three cardinal invariants which are related to the permutation group on the set of natural numbers:ag≔ the least cardinal number of maximal cofinitary permutation groups;ap≔ the least cardinal number of maximal almost disjoint permutation families;c(Sym(N))≔ the cofinality of the permutation group on the set of natural numbers.We show that it is consistent with ZFC that ap=ag<c(Sym(N))=ℵ2; in fact we show that in the Miller model ap=ag=ℵ1<ℵ2=c(Sym(N))

Rigidity of Corona Algebras

In this thesis we use techniques from set theory and model theory to study the isomorphisms between certain classes of C*-algebras. In particular we look at the isomorphisms between corona algebras of the form $\prod\mathbb{M}_{k(n)}(\mathbb{C})/\bigoplus \mathbb{M}_{k(n)}(\mathbb{C})$ for sequences of natural numbers $\{k(n): n\in\mathbb{N}\}$. We will show that the question ``whether any isomorphism between these C*-algebras is trivial", is independent from the usual axioms of set theory (ZFC). We extend the classical Feferman-Vaught theorem to reduced products of metric structures. This implies that the reduced powers of elementarily equivalent structures are elementarily equivalent. We also use this to find examples of corona algebras of the form $\prod\mathbb{M}_{k(n)}(\mathbb{C}) / \bigoplus \mathbb{M}_{k(n)}(\mathbb{C})$ which are non-trivially isomorphic under the Continuum Hypothesis. This gives the first example of genuinely non-commutative structures with this property. In chapter 6 we show that $SAW^{*}$-algebras are not isomorphic to $\nu$-tensor products of two infinite dimensional C*-algebras, for any C*-norm $\nu$. This answers a question of S. Wassermann who asked whether the Calkin algebra has this property