Local actions of P_N, the group of finite permutations on N, on quasi-local algebras are defined and proved to be P_N-abelian. 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 P_N 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.

Local actions of PN, the group of finite permutations on N, on quasi-local algebras are defined and proved to be PN-abelian. 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.

### 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 PN-abelian. 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.
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2023
Local actions of P_N, the group of finite permutations on N, on quasi-local algebras are defined and proved to be P_N-abelian. 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 P_N 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11586/417110`
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