标题： 几乎几乎周期型$\mathrm{III}_1$因子及其3上同调障碍 显示英文标题

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

摘要： 我们构造了一个满因子$M$的示例，使其规范外模流$\sigma^M : \mathbb{R} \rightarrow \mathrm{Out}(M)$是几乎周期性的，但$M$没有几乎周期性状态。 这只有在$\sigma^M$的离散谱中包含一个非平凡的整数二次关系时才可能发生。 我们展示了这种非平凡关系如何产生对几乎周期性状态存在的3上同调障碍。 为了得到我们的主要定理，我们首先加强了Bischoff和Karmakar最近的结果，表明对于任何紧致连通阿贝尔群$K$，$ H^3(K,\mathbb{T})$中的每个上同调类都可以作为超有限$\mathrm{II}_1$因子上的$K$-核的障碍来实现。 我们还证明了一个正面结果：如果对于一个满因子$M$，外模流$\sigma^M : \mathbb{R} \rightarrow \mathrm{Out}(M)$是几乎周期的，那么$M \otimes R$有一个几乎周期态，其中$R$是超有限的$\mathrm{II}_1$因子。 最后，我们证明了关于双曲群强遍历作用相关的交叉积因子的一个正面结果。 显示英文摘要

摘要： 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.