论文标题
统一的理想和商的半群
Unitary Cuntz semigroups of ideals and quotients
论文作者
论文摘要
我们为在[3]中引入并称为Cu $^\ sim $的抽象统一Cuntz半群中定义了理想的概念。我们表明，Cu $^\ sim $ -semigroup的一组理想具有完整的晶格结构。实际上，我们证明，对于任何C $^*$ - 稳定等级的代数一$ a $，分配$ i \ longmapsto $ cu $ _1（i）$定义了一组$ $ a $的完整晶格同构和其单独的cuuntz semigroup cu $ _1（a a）$。此外，我们介绍了（非Abelian）类别Cu $^\ sim $的商和精确性的概念。我们表明，对于任何理想的$ i $ _1（a/i）$，cu $ _1（a）/$ cu $ _1（i）\ simeq $ cu $ _1（a/i）$对于任何理想的$ i $ in $ a $ in $ a $ and $ a $ and $ _1 $ _1 $ _1 $是准确的。最后，我们将cu $^\ sim $ -semigroup与其正元素的cu-序列和其最大元素的abelian组相连。该结果使我们可以提取稳定等级的C $^*$的统一Cuntz Semigroup中的其他信息。
We define a notion of ideal for objects in the category of abstract unitary Cuntz semigroups introduced in [3] and termed Cu$^\sim$. We show that the set of ideals of a Cu$^\sim$-semigroup has a complete lattice structure. In fact, we prove that for any C$^*$-algebra of stable rank one $A$, the assignment $I\longmapsto$Cu$_1(I)$ defines a complete lattice isomorphism between the set of ideals of $A$ and the set of ideals of its unitary Cuntz semigroup Cu$_1(A)$. Further, we introduce a notion of quotients and exactness for the (non abelian) category Cu$^\sim$. We show that Cu$_1(A)/$Cu$_1(I)\simeq$ Cu$_1(A/I)$ for any ideal $I$ in $A$ and that the functor Cu$_1$ is exact. Finally, we link a Cu$^\sim$-semigroup with the Cu-semigroup of its positive elements and the abelian group of its maximal elements in a split-exact sequence. This result allows us to extract additional information that lies within the unitary Cuntz semigroup of a C$^*$-algebra of stable rank one.
