Inclusions of simple C$^*$-algebras arising from compact group actions

Inclusions of operator algebras have long been studied. In particular, inclusions arising from actions of compact groups on factors were studied by Izumi-Longo-Popa and others. The correspondence between intermediate subfactors and subgroups is called the Galois correspondence. Analogues for actions on C$^*$-algebras have been studied by Izumi, Cameron-Smith, Peligrad, and others. In this article, we show the Galois correspondence for quasi-product actions of compact groups on C*-algebras. The notion of a quasi-product action was introduced by Bratteli-Elliott-Kishimoto. Recently, Izumi proved that every minimal action with a simple fixed-point algebra is a quasi-product action. According to these results, we get the Galois correspondence for minimal actions of compact groups with separable simple fixed point algebras. In addition, this paper provides another proof of the Galois correspondence for isometrically shift-absorbing actions and free product actions.