自由超图C* 代数的奇异性质
Curious properties of free hypergraph C*-algebras
摘要：
一个有限超图$H$由一个有限的顶点集$V(H)$和一个子集集$E(H) \subseteq 2^{V(H)}$组成，我们认为这些子集是投影算子之间单位关系的划分。这些单位关系的划分自由地生成一个泛C*-代数，我们称之为“自由超图C*-代数“$C^*(H)$ .一般自由超图C* 代数首先在量子语境下得到研究。作为特殊情况，自由超图C* 代数-代数类包括量子置换群、有限循环群的图积的极大群C* 代数-代数以及与量子图同态、同构和着色有关的C* 代数-代数。本文首次对自由超图C* 代数的各个方面进行了系统的研究。证明了它们与有限维交换C* 代数-代数的有限余极限类以及与同步非局部对策相关的C* 代数-代数的有限余极限类是一致的.我们以前已经证明，它是不可判定的，以确定是否$C^*(H)$是非零给定的$H$。我们现在证明，确定给定的$C^*(H)$是否是剩余有限维的也是不可判定的，同样地，它是否只有无限维表示，以及它是否有迹态。因此，对于这些性质中的每一个，都存在$H$，使得$C^*(H)$是否具有该性质的问题与ZFC公理无关，假设这些公理是一致的。我们在附录中澄清了与这种独立性结果相关的一些微妙之处。
英文摘要：
A finite hypergraph $H$ consists of a finite set of vertices $V(H)$ and a collection of subsets $E(H) \subseteq 2^{V(H)}$ which we consider as partition of unity relations between projection operators. These partition of unity relations freely generate a universal C*-algebra, which we call the "free hypergraph C*-algebra" $C^*(H)$. General free hypergraph C*-algebras were first studied in the context of quantum contextuality. As special cases, the class of free hypergraph C*-algebras comprises quantum permutation groups, maximal group C*-algebras of graph products of finite cyclic groups, and the C*-algebras associated to quantum graph homomorphism, isomorphism, and colouring. Here, we conduct the first systematic study of aspects of free hypergraph C*-algebras. We show that they coincide with the class of finite colimits of finite-dimensional commutative C*-algebras, and also with the class of C*-algebras associated to synchronous nonlocal games. We had previously shown that it is undecidable to determine whether $C^*(H)$ is nonzero for given $H$. We now show that it is also undecidable to determine whether a given $C^*(H)$ is residually finite-dimensional, and similarly whether it only has infinite-dimensional representations, and whether it has a tracial state. It follows that for each one of these properties, there is $H$ such that the question whether $C^*(H)$ has this property is independent of the ZFC axioms, assuming that these are consistent. We clarify some of the subtleties associated with such independence results in an appendix.
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