标题： 在分次的$C^*$-代数中提升$K$-理论中的同构 显示英文标题

标题： Lifting isomorphisms in $K$-theory through gradings of $C^*$-algebras

摘要： 我们证明每个强$\mathbb{Z}$-分次 C*-代数（等价地，每个带有满谱子空间的强连续$\mathbb{T}$-作用的 C*-代数）都是 Cuntz--Pimsner 代数，并描述了可用于构造中的系数代数和模的子代数和子空间。 我们推导出对于C*-代数$A$的满射分次同态$\phi$，这些C*-代数由无扭阿贝尔群$H$进行分次，如果$\phi$在$A$的零分次部分$A_0$上的限制$\phi_0$在K理论中诱导同构，那么$\phi$本身也是如此。 当$H$是自由阿贝尔时，我们说明如何选取$A_0$的更小子代数，在这些子代数上只需检查$\phi$在$K$-理论中是否诱导同构。 显示英文摘要

摘要： We show that every strongly $\mathbb{Z}$-graded C*-algebra (equivalently, every C*-algebra carrying a strongly continuous $\mathbb{T}$-action with full spectral subspaces) is a Cuntz--Pimsner algebra, and describe subalgebras and subspaces that can be used as the coefficient algebra and module in the construction. We deduce that for surjective graded homomorphisms $\phi$ of C*-algebras $A$ graded by torsion-free abelian groups $H$, if the restriction $\phi_0$ of $\phi$ to the zero-graded component $A_0$ of $A$ induces isomorphisms in K-theory, so does $\phi$ itself. When $H$ is free abelian, we show how to pick out smaller subalgebras of $A_0$ on which it suffices to check that $\phi$ induces isomorphisms in $K$-theory.