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

Abstract

We construct an exemple of a full factor $M$ such that its canonical outer modular flow ${\sigma }^M:\mathbb{R}\to \mathrm{Out}(M)$ 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 ${\sigma }^M:\mathbb{R}\to \mathrm{Out}(M)$ 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.

29 pages