De finetti-type theorems on quasi-local algebras and infinite fermi tensor products

Abstract

Local actions of PN, the group of finite permutations on N, on quasi-local algebras are defined and proved to be PNabelian. It turns out that invariant states under local actions are automatically even, and extreme invariant states are strongly clustering. Tail algebras of invariant states are shown to obey a form of the Hewitt and Savage theorem, in that they coincide with the fixed-point von Neumann algebra. Infinite graded tensor products of C∗-algebras, which include the CAR algebra, are then addressed as particular examples of quasi-local algebras acted upon PN in a natural way. Extreme invariant states are characterized as infinite products of a single even state, and a de Finetti theorem is established. Finally, infinite products of factorial even states are shown to be factorial by applying a twisted version of the tensor product commutation theorem, which is also derived here. Mathematics Subject Classification: 46L06, 60G09, 60F20, 46L53.