Spectral gap and strict outerness for actions of locally compact groups on full factors

TL;DR

This paper establishes that outer actions of locally compact groups on full factors are automatically strictly outer and that the crossed product remains full if the group image is closed in the outer automorphism group, using spectral gap techniques.

Contribution

It extends known results from discrete and compact groups to locally compact groups, proving strict outerness and fullness preservation via spectral gap methods.

Findings

Outer actions of locally compact groups are strictly outer.

Crossed products remain full under certain conditions.

Spectral gap property is key to these results.

Abstract

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.