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

Abstract: In the present note, which is the second part of a work concerning the study of the set of the symmetric states, we introduce the extension of the Klein transformation for general Fermi tensor product of two $\bz^2$ graded $C^*$-algebras, under the condition that the grading of one of the involved algebras is inner. After extending the construction to $C^*$-inductive limits, such a Klein transformation realises a canonical $$-isomorphism between two $\bz^2$-graded $C^$-algebras made of the infinite Fermi $C^*$-tensor product and the infinite $C^*$-tensor product of a single $\bz^2$-graded $C^*$-algebra, both built with respect to the corresponding minimal $C^*$-cross norms. It preserves the grading, and its transpose sends even product states of $\ga_{\rm X}$ in (necessarily even) product states on $\ga_{\rm F}$, and therefore induces an isomorphism of simplexes $$ \cs_\bp(\ga_{\rm F})=\cs_{\bp\times\bz^2}(\ga_{\rm F})\sim\cs_{\bp\times\bz^2}(\ga_{\rm X})\,, $$ which allows to reduce the study of the structure of the symmetric states for $C^*$-Fermi systems to the corresponding even symmetric states on the usual infinite $C^*$-tensor product. Other relevant properties of symmetric states on the Fermi algebra will be proved without the use of the Klein transformation. We end with an example for which such a Klein transformation is not implementable, simply because the Fermi tensor product does not generate a usual tensor product. Therefore, in general, the study of the symmetric states on the Fermi algebra cannot be reduced to that of the corresponding symmetric states on the usual infinite tensor product, even if both share many common properties.