$C^*$ -Fermi系统的对称态II：Klein变换及其结构
Symmetric states for $C^*$-Fermi systems II: Klein transformation and their structure
摘要：
本文是关于对称态集研究的第二部分，在其中一个$\bz^2$分次$C^*$ -代数的分次为内的条件下，引入了两个$\bz^2$分次$C^*$ -代数的一般Fermi张量积的Klein变换的扩张.在将构造扩展到$C^*$ -归纳极限之后，这样的Klein变换实现了两个$\bz^2$ -分次$C^*$ -代数之间的规范$*$ -同构，这两个分次$C^*$ -代数由单个$\bz^2$ -分次$C^*$ -代数的无限费米$C^*$ -张量积和无限$C^*$ -张量积构成，这两个分次$C^*$ -代数都是关于相应的最小$C^*$ -交叉范数建立的.它保持了分级，并且它的转置将$\ga_{\rm X}$的偶数乘积态发送到$\ga_{\rm F}$上的（必然是偶数）乘积态，因此诱导了单形$$ \cs_\bp(\ga_{\rm F})=\cs_{\bp\times\bz^2}(\ga_{\rm F})\sim\cs_{\bp\times\bz^2}(\ga_{\rm X})\,, $$的同构，这允许将$C^*$ -费米系统的对称态结构的研究减少到通常的无限$C^*$ -张量积上的相应偶数对称态。费米代数上对称态的其他相关性质将在不使用克莱因变换的情况下得到证明。最后，我们举一个例子，对于这个例子来说，克莱因变换是不可实现的，因为费米张量积不能产生通常的张量积。因此，一般来说，对费米代数上对称态的研究不能归结为对通常的无限张量积上相应对称态的研究，即使两者有许多共同的性质。
英文摘要：
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.
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