量子CW复形在单位C*-代数的Waldhausen范畴中
Quantum CW-complexes in a Waldhausen category for unital C*-algebras
AI总结：
本文在Waldhausen范畴框架下定义有限量子CW复形，通过添加K-理论同构的逆来丰富态射，并利用图代数方法将环结构从经典复射影空间传输到量子版本，从而简化多推出量子复射影空间的K-理论计算。
摘要：
利用有限CW复形的K-群的环结构，Atiyah和Todd揭示了复射影空间$CP^n$的K-理论。在此，在非交换拓扑的领域中，我们使用Waldhausen范畴的语言发展了一个新的有限量子CW复形框架，该框架允许我们通过添加在K-理论中为同构的*-同态的逆来丰富单位C*-代数之间的标准态射类。我们的量子CW复形概念包含了早期构造，并拥有众多例子。此外，该框架使我们能够将关于多推出量子复射影空间$CP^n_H$的问题简化为更易处理的Vaksman-Soibelman量子复射影空间$CP^n_q$的设置，后者可利用图代数方法。特别是，这些方法使我们能够将环结构从$K^0(CP^n)$传输到$K^0(CP_q^n)$。最后，我们调整Waldhausen范畴的形式，以从$K^0(CP_q^n)$的自然自由生成元集确定$K^0(CP_H^n)$的自然自由生成元集，并将环结构从$K^0(CP_q^n)$传输到$K^0(CP_H^n)$。
英文摘要：
Using the ring structure of the K-groups of finite CW-complexes, Atiyah and Todd unravelled the K-theory of complex projective spaces $CP^n$. Herein, in the realm of noncommutative topology, we develop a new framework of finite quantum CW-complexes using the language of Waldhausen categories, which allows us to enrich the class of standard morphisms between unital C*-algebras by adding inverses of *-homomorphisms that are isomorphisms in K-theory. Our concept of quantum CW-complexes subsumes earlier constructions and enjoys a plethora of examples. Moreover, the framework allows us to reduce problems concerning the multipushout quantum complex projective space $CP^n_H$ to the much more approachable setting of the Vaksman-Soibelman quantum complex projective space $CP^n_q$ enjoying the availability of graph-algebraic methods. In particular, these methods permit us to transport the ring structure from $K^0(CP^n)$ to $K^0(CP_q^n)$. Finally, we adapt the formalism of Waldhausen categories to determine a natural set of free generators of $K^0(CP_H^n)$ from a natural set of free generators of $K^0(CP_q^n)$, and to transport the ring structure from $K^0(CP_q^n)$ to $K^0(CP_H^n)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
