局部紧群在全因子上作用的谱隙与严格外性
Spectral gap and strict outerness for actions of locally compact groups on full factors
AI总结：
本研究证明局部紧群在全因子上的外作用自动严格外且对应包含具谱隙，群像闭时交叉积仍为全因子，填补了离散与紧群之外的结果空白。
摘要：
我们证明，局部紧群$G$在全因子$M$上的外作用自动是严格外的，即$M$在交叉积中的相对换位子是平凡的。若进一步$G$在外自同构群$\text{Out }M$中的像为闭集，我们证明交叉积仍为全因子。我们通过证明$M$到交叉积的包含自动具有谱隙性质得到该结果。此前这类结果仅对离散群作用和紧群作用被证明，且两类情况使用了截然不同的方法。即便对于自由群因子或自由Araki-Woods因子上的典范Bogoljubov作用，这些结果也都是新的。
英文摘要：
We prove that an outer action of a locally compact group $G$ on a full factor $M$ is automatically strictly outer, meaning that the relative commutant of $M$ in the crossed product is trivial. If moreover the image of $G$ in the outer automorphism group $\operatorname{Out} M$ is closed, we prove that the crossed product remains full. We obtain this result by proving that the inclusion of $M$ in the crossed product automatically has a spectral gap property. Such results had only been proven for actions of discrete groups and for actions of compact groups, by using quite different methods in both cases. Even for the canonical Bogoljubov actions on free group factors or free Araki-Woods factors, these results are new.
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