In functional analysis, approximative properties of an object become precise in its ultrapower. We discuss this idea and its consequences for automorphisms of II1 factors. Here are some sample results: (1) an automorphism is approximately inner if and only if its ultrapower is א0-locally inner; (2) the ultrapower of an outer automorphism is always outer; (3) for unital *-homomorphisms from a se...

We study when certain properties of Banach algebras are stable under ultrapower constructions. In particular, we consider when every ultrapower of A is Arens regular, and give some evidence that this is so if and only if A is isomorphic to a closed subalgebra of operators on a super-reflexive Banach space. We show that such ideas are closely related to whether one can sensibly define an ultrapo...

We define higher order fixed points of normal functions, describe them and apply to obtain a constructive proof that, if κ is the least ordinal such that the ultrapower κ/F is non-trivial, then that ultrapower has at least κ elements. AMS Mathematics Subject Classification (2000): 03E04, 03E20, 03C20.

I prove that there is a recursive function T that does the following: Let X be transitive and rud closed, and let X ′ be the closure of X ∪{X} under rud functions. Given a Σ0 formula φ(x) and a code c for a rud function f , T (φ, c, ~x) is a Σω formula such that for any ~a ∈ X, X ′ |= φ[f(~a)] iff X |= T (φ, c, ~x)[~a]. I make this precise and show relativized versions of this. As an applicatio...

We show that while the length ω iterated ultrapower by a normal ultrafilter is a Boolean ultrapower by the Boolean algebra of Př́ıkrý forcing, it is consistent that no iteration of length greater than ω (of the same ultrafilter and its images) is a Boolean ultrapower. For longer iterations, where different ultrafilters are used, this is possible, though, and we give Magidor forcing and a general...

We use continuous model theory to obtain several results concerning isomorphisms and embeddings between II1 factors and their ultrapowers. Among other things, we show that for any II1 factor M, there are continuum many nonisomorphic separable II1 factors that have an ultrapower isomorphic to an ultrapower of M. We also give a poor man’s resolution of the Connes Embedding Problem: there exists a...

A construction of the real number system based on almost homomorphisms of the integers Z was proposed by Schanuel, Arthan, and others. We combine such a construction with the ultrapower or limit ultrapower construction, to construct the hyperreals out of integers. In fact, any hyperreal field, whose universe is a set, can be obtained by such a one-step construction directly out of integers. Eve...

We show how we can apply ultrapower methods to density problems in additive/combinatorial number theory.

In this paper we propose a new approach to realizability interpretations for nonstandard arithmetic. We deal with analysis in the context of (semi)intuitionistic realizability, focusing on Lightstone-Robinson construction model through an ultrapower. particular, consider extension $\lambda$-calculus memory cell, that contains integer (the state), order indicate which slice ultrapower $\cal{M}^{...