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Local actions of $\mathbb{P}_\mathbb{N}$, the group of finite permutations on$\mathbb{N}$, on quasi-local algebras are defined and proved to be$\mathbb{P}_\mathbb{N}$-abelian. It turns out that invariant states under localactions are automatically even, and extreme invariant states are stronglyclustering. Tail algebras of invariant states are shown to obey a form of theHewitt and Savage theorem, in that they coincide with the fixed-point vonNeumann algebra. Infinite graded tensor products of $C^*$-algebras, whichinclude the CAR algebra, are then addressed as particular examples ofquasi-local algebras acted upon $\mathbb{P}_\mathbb{N}$ in a natural way.Extreme invariant states are characterized as infinite products of a singleeven state, and a de Finetti theorem is established. Finally, infinite productsof factorial even states are shown to be factorial by applying a twistedversion of the tensor product commutation theorem, which is also derived here.

de Finetti-type theorems on quasi-local algebras and infinite Fermi tensor products