Almost almost periodic type $\mathrm{III}_1$ factors and their 3-cohomology obstructions

TL;DR

This paper constructs examples of full factors with almost periodic outer modular flows lacking almost periodic states, revealing a 3-cohomology obstruction linked to nontrivial quadratic relations in the spectrum.

Contribution

It demonstrates how 3-cohomology obstructions can prevent the existence of almost periodic states in certain full factors, extending previous results to new classes of factors.

Findings

Construction of full factors with almost periodic outer modular flow but no almost periodic state

Realization of any cohomology class in $H^3(K, ext{T})$ as an obstruction on the hyperfinite II_1 factor

Proved that tensoring with the hyperfinite II_1 factor yields an almost periodic state under certain conditions

Abstract

We construct an exemple of a full factor $M$ such that its canonical outer modular flow $\sigma^M : \mathbb{R} \rightarrow \mathrm{Out}(M)$ is almost periodic but $M$ has no almost periodic state. This can only happen if the discrete spectrum of $\sigma^M$ contains a nontrivial integral quadratic relation. We show how such a nontrivial relation can produce a 3-cohomological obstruction to the existence of an almost periodic state. To obtain our main theorem, we first strengthen a recent result of Bischoff and Karmakar by showing that for any compact connected abelian group $K$, every cohomology class in $ H^3(K,\mathbb{T})$ can be realized as an obstruction of a $K$-kernel on the hyperfinite $\mathrm{II}_1$ factor. We also prove a positive result : if for a full factor $M$ the outer modular flow $\sigma^M : \mathbb{R} \rightarrow \mathrm{Out}(M)$ is almost periodic, then $M \otimes R$ has an almost periodic state, where $R$ is the hyperfinite $\mathrm{II}_1$ factor. Finally, we prove a positive result for crossed product factors associated to strongly ergodic actions of hyperbolic groups.