  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.
