We show that there are II1 factors M and elementary embeddings M-MU which do not lift to sequences of ucp maps, and in fact M can be chosen from any given elementary equivalence class. Furthermore, under continuum hypothesis, we show that in the sense of cardinality "most" automorphisms of an ultrapower MU of a separable II1 factor do not lift to a sequence of ucp maps phi n : M-M.
In classical model theory, the Keisler-Shelah theorem establishes a fundamental connection between the elementary equivalence of structures and the isomorphism of their ultrapowers. Motivated by this, one may ask whether an analogous relationship holds in the framework of continuous model theory, which naturally encompasses metric structures such as C*-algebras. In this paper, we investigate the isomorphism problem for ultraproducts of operator algebras from a model-theoretic perspective. We prove that, assuming the negation of the continuum hypothesis, there exist two elementarily equivalent infinite-dimensional unital C*-algebras A and B, whose density characters are at most c, such that for all non-principal ultrafilters U, V on omega, the ultrapowers AU and BV are not isomorphic. This result provides a continuous analogue of certain classical theorems concerning ultraproducts and demonstrates that the model-theoretic behavior of C*-algebras is closely related to set-theoretic principles such as the continuum hypothesis.
We compare the approaches of Schr & ouml;der and Weihrauch to computable topology. These approaches are both based on the theory of representations, but they focus on a priori different representations of open sets. We show that the two approaches are compatible. While it is well known that Schr & ouml;der's approach is more general because it can be applied to non-second countable spaces, we insist on the fact that it is also more general even when dealing only with second countable spaces, because a represented space can be second countable without being effectively second countable. represented spaces that embed into computable metric spaces: those are the computably second countable strongly computably regular represented spaces. Finally, we study different forms of open choice problems. We show that having a computable open choice is equivalent to being computably separable, but that the "non-total open choice problem", i.e., open choice restricted to open sets that have non-empty complement, interacts with effective second countability in a satisfying way.
We give a shorter proof of a theorem of G. Elek stating that two hyperfinite measure-preserving actions of a countable group on standard probability spaces are approximately conjugate if and only if they have the same invariant random subgroup. We then use this theorem to study model theory of hyperfinite measure-preserving actions of countable groups on probability spaces. This work generalizes the model-theoretic study of automorphisms of probability spaces conducted by I. Ben Yaacov, A. Berenstein, C. W. Henson and A. Usvyatsov.
