We introduce a new invariant for $C^{⁎}$-algebras of stable rank one that merges the Cuntz semigroup information together with the ${\mathrm{K}}_1$-group information. This semigroup, termed the ${\mathrm{Cu}}_1$-semigroup, is constructed as equivalence classes of pairs consisting of a positive element in the stabilization of the given $C^{⁎}$-algebra together with a unitary element of the unitization of the hereditary subalgebra generated by the given positive element. We show that the ${\mathrm{Cu}}_1$-semigroup is a well-defined continuous functor from the category of $C^{⁎}$-algebras of stable rank one to a suitable codomain category that we write ${\mathrm{Cu}}^{\sim }$. Furthermore, we compute the ${\mathrm{Cu}}_1$-semigroup of some specific classes of $C^{⁎}$-algebras. Finally, in the course of our investigation, we show that we can recover functorially Cu, ${\mathrm{K}}_1$ and ${\mathrm{K}}_{⁎}:={\mathrm{K}}_0\oplus {\mathrm{K}}_1$ from ${\mathrm{Cu}}_1$.

我们引入了一个新的不变量 $C^{⁎}$-将 Cuntz 半群信息与 ${\mathrm{钾}}_1$- 组信息。这个半群称为${\mathrm{铜}}_1$-semigroup，被构造为由稳定给定的正元素组成的对的等价类 $C^{⁎}$-algebra 连同由给定正元素生成的遗传子代数的单元化的酉元素。我们表明${\mathrm{铜}}_1$-semigroup 是一个定义良好的连续函子，来自 $C^{⁎}$-稳定排名第一的代数到我们写的合适的codomain类别 ${\mathrm{铜}}^{～}$. 此外，我们计算${\mathrm{铜}}_1$- 某些特定类别的半群 $C^{⁎}$-代数。最后，在我们的调查过程中，我们表明我们可以在函数上恢复 Cu，${\mathrm{钾}}_1$ 和 ${\mathrm{钾}}_{⁎}：={\mathrm{钾}}_0\oplus {\mathrm{钾}}_1$ 从 ${\mathrm{铜}}_1$.