Conditional expectations associated with strongly quasi-invariant states and an application to spin systems

We discuss the conditional expectations and martingales in relevance with G-strongly quasi-invariant states on a C*-algebra A , where G is a separable locally compact group of ∗-automorphisms of A . In the von Neumann algebra A of the GNS representation, we define a unitary representation of the group and a group G ̂ of ∗-automorphisms of A , which is homomorphic to G. For the case of compact G, we find a G ̂ -invariant state on A and define a conditional expectation with range the G ̂ -fixed subalgebra. When G is the union of increasing compact groups, we construct a sequence of conditional expectations and thereby construct (backward) martingales, which have limits by the martingale convergence theorem. As an example we consider S∞ the group of local permutations which acts on a C*-algebra of infinite tensor product of finite dimensional C*-algebras. We also find an application in classical spin systems.