Let $G$ be a finite group. I have an action of $G$ on a matrix algebra of positive operators, $\mathcal{M}$. In particular, $\mathcal{M}$ has a $G$-module structure, yielding a linear representation of $G$. Let $\mathcal{C}$ denote the commutant of the representation.
Assume a positive operator, $T$, is given. I would ideally like to define a multiplicative average of $T$, along the lines of the geometric of positive scalars, leveraging the matrix multiplication structure on the algebra. The average should ideally be a projection of $T$ on the commutant of the representation. Ultimately, I would like to generalize this to the case of compact Lie groups as well.
Can someone point me towards suitable references? I would like a systematic approach based on perhaps the geometry of the manifold of positive operators to define the suitable operation above.