论文标题
$ C^*$ - 费米系统II：Klein转换及其结构的对称状态
Symmetric states for $C^*$-Fermi systems II: Klein transformation and their structure
论文作者
论文摘要
在本说明中，这是有关对称状态的研究的第二部分，我们介绍了两个$ \ bz^2 $分级$ c^*$ - 代数的klein转换的扩展，并在所涉及的代数分级的条件下是内在的。在将施工扩展到$ c^*$ - 归纳限制之后，这种klein转换实现了一个规范的$*$ - 同构 - 在两个$ \ bz^2 $ - 级别的$ c^*$ - 由Infinite fermi $ c^*$ c^*$ tensor和Infinite $ c^*$ c^*$ - $ -C^*$ - $ -C^*$ - tensor产品^$ c^*$ - tensor产品的代数$ c^*$ - 代数，均相对于相应的最小$ c^*$ - 交叉标准构建。它保留了分级，其转置在$ \ ga _ {\ rm f} $上以$ \ ga _ {\ rm x} $的产品状态发送均匀的产品状态，从而引起了单纯质量的同构，因此诱发了单纯态的同构\ cs_ \ cs_ \ bp（\ bp（\ bp）（\ bp） f}）= \ cs _ {\ bp \ times \ bz^2}（\ ga _ {\ rm f}）\ sim \ sim \ cs _ {\ bp \ bp \ bp \ bp \ bp \ bp \ bz^2}（\ ga _ _ _ _ {\ rm x}} \，$ cysertir的$ csysect for-c syserce for-c sysent for-c sysercement-for-在通常的无限$ C^*$ - 张量产品上的相应的对称状态。在不使用klein转换的情况下，将证明对称状态在费米代数上的其他相关特性。 我们以一个示例结尾，这样的klein转换无法实现，仅仅是因为费米张量产品不会生成通常的张量产品。因此，通常，即使两者具有许多共同特性，也不能将对Fermi代数对称状态的研究降低到通常的无限张量产物上的相应对称状态。
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.
