Fixed point algebra

Question

Suppose that $$ is a von Neuman algebra and we have an action of a finite group $G$ on $M$. Denote by $M^G$ the fixed point subalgebra and suppose that $M^G=\mathbb{C}$ (i.e., we have an ergodic action of $G$ on $M$). I am looking for an elementary way to see why $M$ is finite dimensional. Any help/hint would be appreciated.