Symmetric states for $C^*$-Fermi systems I: De Finetti theorem

Abstract

In the present note, which is the first part of a work concerning the study of the set of the symmetric states for Fermi systems, we describe the extension of the De Finetti theorem to the infinite Fermi C * -tensor product of a single (separable) general Z 2 -graded C * -algebra.

Symmetric states for $C^*$-Fermi systems II: Klein transformation and their structure

“…We gather here some facts useful for the forthcoming sections, and refer the reader to Section 2 of [12] for the remaining notations and basic results frequently used here.…”

“…By coming back to our study, in [12] we indeed built the infinite C * -Fermi tensor product F N B, F N α of the single Z 2 -graded C *algebra B by using such a "minimal" norm. Then we studied the early properties of symmetric states, that is the states in the simplex S P (A F ) consisting of all states invariant under the natural action of the group P N =: P of all finite permutations of N, on A F .…”

“…The key point to construct such a framework is the GNS representation of the product state ω × ϕ on A F B for even states ω ∈ S + (A) and ϕ ∈ S + (B). Although the detailed knowledge of such a GNS construction plays no role in the analysis in [12], it is relevant here to prove that the Klein transformation we are building in the foregoing section is realising a * -isomorphism between the Fermi tensor product and the usual tensor product. Such a detailed analysis is carried out in [8], Section 3, which we report here for the convenience of the reader.…”

“…together with the infinite product state × N ϕ ∈ S + (A F ) of the single state ϕ ∈ S + (B). The reader is referred to [12] for further details.…”

“…The present note is the second part relative to the investigation of the main properties of the so called symmetric states on Fermi systems. Indeed, to be more precise, in the 1st part [12], we built the infinite C * -Fermi tensor product (A F , α) := F N B, F N β of the single Z 2graded C * -algebra B by using the analogous of the minimal norm of the usual C * -tensor product firstly introduced at the end of Section 9 of [7].…”