论文标题
量子Cuntz-Krieger代数
Quantum Cuntz-Krieger algebras
论文作者
论文摘要
由Cuntz-Krieger代数理论的动机，我们定义并研究了与有向量子图相关的$ C^\ ast $ - 代数。对于古典图，以这种方式获得的$ c^\ ast $ algebras可以看作是Cuntz-Krieger代数的免费类似物，并且不必是核。 我们详细研究了两种特定类别的量子图，即微不足道和完整的量子图。对于单个矩阵块上的琐碎量子图，我们表明相关的量子cuntz-krieger代数既不是Unital，核也不简单，也不依赖于矩阵块的大小高达$ kk $ - $ - 等价性。在完整的量子图的情况下，我们使用量子对称性来表明，在某些情况下，相应的量子cuntz-krieger代数对Cuntz代数是同构的。这些同构似乎远非定义显而易见，特别是暗示着这些$ c^\ ast $ - 代理人都是成对的非同构，即使在$ kk $ theory的级别上，对于不同维度的完整量子图。我们解释了量子信息理论中统一错误基础的概念如何有助于阐明情况。 我们还讨论了量子cuntz-krieger代数的量子对称性。
Motivated by the theory of Cuntz-Krieger algebras we define and study $ C^\ast $-algebras associated to directed quantum graphs. For classical graphs the $ C^\ast $-algebras obtained this way can be viewed as free analogues of Cuntz-Krieger algebras, and need not be nuclear. We study two particular classes of quantum graphs in detail, namely the trivial and the complete quantum graphs. For the trivial quantum graph on a single matrix block, we show that the associated quantum Cuntz-Krieger algebra is neither unital, nuclear nor simple, and does not depend on the size of the matrix block up to $ KK $-equivalence. In the case of the complete quantum graphs we use quantum symmetries to show that, in certain cases, the corresponding quantum Cuntz-Krieger algebras are isomorphic to Cuntz algebras. These isomorphisms, which seem far from obvious from the definitions, imply in particular that these $ C^\ast $-algebras are all pairwise non-isomorphic for complete quantum graphs of different dimensions, even on the level of $ KK $-theory. We explain how the notion of unitary error basis from quantum information theory can help to elucidate the situation. We also discuss quantum symmetries of quantum Cuntz-Krieger algebras in general.
