We prove a strong dichotomy for the number of ultrapowers of a given model of cardinality ≤ 2ℵ0 associated with nonprincipal ultrafilters on ℕ. They are either all isomorphic, or else there are 22ℵ0 many nonisomorphic ultrapowers. We prove the analogous result for metric structures, including C*-algebras and II1 factors, as well as their relative commutants and include several applications. We also show that the CAF001-algebra [Formula: see text] always has nonisomorphic relative commutants in its ultrapowers associated with nonprincipal ultrafilters (...) on ℕ. (shrink)
We consider various quotients of the C*-algebra of bounded operators on a nonseparable Hilbert space, and prove in some cases that, assuming some restriction of the Generalized Continuum Hypothesis, there are many outer automorphisms.
The purpose of this communication is to survey a theory of liftings, as developed in author's thesis. The first result in this area was Shelah's construction of a model of set theory in which every automorphism of P/ Fin, where Fin is the ideal of finite sets, is trivial, or inother words, it is induced by a function mapping integers into integers. Soon afterwards, Velickovic, was able to extract from Shelah's argument the fact that every automorphism of P/ Fin with (...) a Baire-measurable lifting has to be trivial. This, for instance, implies that in Solovay's model all automorphisms are trivial. Later on, an axiomatic approach was adopted and Shelah's conclusion was drawn first from the Proper Forcing Axiom and then from the milder Open Coloring Axiom and Martin's Axiom. Both shifts from the quotient P/ Fin to quotients over more general ideals P/I and from automorphisms to arbitrary ho-momorphisms were made by Just in a series of papers, motivated by some problems in algebra and topology. (shrink)
We use modified Tsirelson's spaces to prove that there is no finite basis for turbulent Polish group actions. This answers a question of Hjorth and Kechris 329–346; Hjorth, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2000, Section 3.4.3).