  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 give examples of compact group actions on simple
C$^*$-algebras for which the Galois correspondence holds.
