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

TL;DR

The paper analyzes when a full factor $M$ with almost periodic outer modular flow $\sigma^M$ must admit an almost periodic state. It develops a cohomological obstruction theory for compact connected abelian kernels, linking the 3-cohomology group $H^3(K,\mathbf{T})$ to integral quadratic forms via an isomorphism $\alpha:\mathfrak{Q}_{\rm int}(K)\to H^3(K,\mathbf{T})$, and shows every obstruction class can be realized by a kernel on a II$_1$ factor (hyperfinite or full). A main dichotomy is established: $\log(\Gamma)$ being quadratically free implies that any full factor with $\sigma^M$ almost periodic actually has an almost periodic weight, while the obstruction $\varrho(M)$, living in $\log(\Gamma)\odot\log(\Gamma)$, controls the existence of such weights; this leads to a construction of full factors with almost periodic outer flows that lack almost periodic states. The results extend to tensor products and crossed products by hyperbolic groups, and the framework is also adapted to nonsingular strongly ergodic equivalence relations, yielding positive answers in that setting. Overall, the work clarifies the subtle interplay between modular dynamics, 3-cohomology obstructions, and almost periodicity in type III factors, providing both obstructions and realizations of prescribed cohomological data.

Abstract

We construct an exemple of a full factor $M$ such that its canonical outer modular flow $σ^M : \mathbb{R} \rightarrow \mathrm{Out}(M)$ is almost periodic but $M$ has no almost periodic state. This can only happen if the discrete spectrum of $σ^M$ 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 $σ^M : \mathbb{R} \rightarrow \mathrm{Out}(M)$ 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.

Theorems & Definitions (118)

• Theorem 2
• Corollary 3
• Theorem 4
• Theorem 5
• Corollary 6
• Theorem 7
• Theorem 1.1
• Proposition 1.2
• Proposition 1.3
• Proposition 1.4