Elementary embeddings into ultrapower $\mathrm{II}_1$ factors without a UCP lift

TL;DR

The paper demonstrates the existence of elementary embeddings of certain $ ext{II}_1$ factors into their ultrapowers that cannot be approximated by sequences of UCP maps, highlighting limitations in lifting properties under various set-theoretic assumptions.

Contribution

It constructs examples of elementary embeddings into ultrapowers of $ ext{II}_1$ factors that do not lift to UCP maps, extending understanding of embedding and lifting phenomena in operator algebras.

Findings

Existence of non-liftable elementary embeddings into ultrapowers.

Most automorphisms of ultrapowers do not lift to UCP maps under continuum hypothesis.

Any elementary equivalence class of $ ext{II}_1$ factors can contain such examples.

Abstract

We show that there are $\mathrm{II}_1$ factors $M$ and elementary embeddings $M \to M^{\mathcal{U}}$ 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 a ultrapower $M^{\mathcal{U}}$ of a separable $\mathrm{II}_1$ factor do not lift to a sequence of UCP maps $\varphi_n: M \to M$.