If a compact group $G$ acts on vN algebra factor $M\subset B(L^{2}(M))$, what would be the index of subfactor $[M^{G}:M]$? KIndly explain the answer.



Index of a particular subfactor
Case $G=\mathbb Z/2$, generated by $g\in G$:


In that case, the action of $G$ on $M$ induces a grading $M=M\_0 \oplus M\_1$ where $M\_0=M^G=\{m\in M: gm=m\}$ and $M\_1=\{m\in M: gm=-m\}$.
Similarly, the action of $G$ on $L^2M$ induces a grading $L^2M=(L^2M)\_0\oplus (L^2M)\_1$. The subspace $(L^2M)\_0$ is isomorphic to $L^2(M^G)$.


And both $(L^2M)\_0$ and $(L^2M)\_1$ are isomorphic as $M^G$-modules.
So
$$
\dim\_{M^G}(L^2M) = \dim\_{M^G}(L^2M)\_0 + \dim\_{M^G}(L^2M)\_1 = 2\cdot \dim\_{M^G}(L^2(M^G))=2.
$$


