Galois correspondences for subfactors related to normal subgroups

Commun. Korean Math. Soc. 2002 Vol. 17, No. 2, 253-260 Printed June 1, 2002

Jung-Rye Lee Daejin University

Abstract : For an outer action $\alpha$ of a finite group $G$ on a factor $M$, it was proved that $H$ is a normal subgroup of $G$ if and only if there exists a finite group $F$ and an outer action $\beta $ of $F$ on the crossed product algebra $M\rtimes_\alpha H$ with $M\rtimes_\alpha G \cong (M\rtimes_\alpha H ) \rtimes _\beta F $. We generalize this to infinite group actions. For an outer action $\alpha$ of a discrete group, we obtain a Galois correspondence for crossed product algebras related to normal subgroups. When $\alpha$ satisfies a certain condition, we also obtain a Galois correspondence for fixed point algebras. Furthermore, for a minimal action $\alpha$ of a compact group $G$ and a closed normal subgroup $H$, we prove $M^G =(M^H)^{\beta (G/H)}$ for a minimal action $\beta$ of $G/H$ on $M^H$.