Classification for the Actions of a Compact Abelian Group on a Semifinite Real $W^*$-Algebra

In this article, we study the actions of groups on real von Neumann algebras. A complete classification is obtained for the actions of arbitrary finite groups on hyperfinite real factors of type II$_1$. Using Takesaki's theorem for real von Neumann algebras, we classify (up to conjugacy) the actions of compact abelian groups on hyperfinite real factor of type II$_\infty$ in terms of cocycle-conjugacy of dual actions.