Martingales associated with strongly quasi-invariant states

We discuss the martingales in relevance with $G$-strongly quasi-invariant states on a $C^*$-algebra $\mathcal A$, where $G$ is a separable locally compact group of $*$-automorphisms of $\mathcal A$. In the von Neumann algebra $\mathfrak A$ of the GNS representation, we define a unitary representation of the group and define a group $\hat G$ of $*$-automorphisms of $\mathfrak A$, which is homomorphic to $G$. For the case of compact $G$, under some mild condition, we find a $\hat G$-invariant state on $\mathfrak A$ and define a conditional expectation with range the $\hat G$-fixed subalgebra. Moving to the separable locally compact group $G=\cup_NG_N$, which is the union of increasing compact groups, we construct a sequence of conditional expectations and thereby construct (decreasing) martingales, which have limits by the martingale convergence theorem. We provide with an example for the group of finite permutations on the set of nonnegative integers acting on a $C^*$-algebra of infinite tensor product.