It is shown that the Cuntz semigroup of a space with dimension at most two, and with second cohomology of its compact subsets equal to zero, is isomorphic to the ordered semigroup of lower semicontinuous functions on the space with values in the natural numbers with the infinity adjoined. This computation is then used to obtain the Cuntz semigroup of all compact surfaces. A converse to the first computation is also proven: if the Cuntz semigroup of a separable C*-algebra is isomorphic to the lower semicontinuous functions on a topological space with values in the extended natural numbers, then the C*-algebra is commutative up to stability, and its spectrum satisfies the dimensional and cohomological conditions mentioned above.
We show that the dimension of the Cuntz semigroup of a C*-algebra is determined by the dimensions of the Cuntz semigroups of its separable sub-C*-algebras. This allows us to remove separability assumptions from previous results on the dimension of Cuntz semigroups. To obtain these results, we introduce a notion of approximation for abstract Cuntz semigroups that is compatible with the approximation of a C*-algebra by sub-C*-algebras. We show that many properties for Cuntz semigroups are preserved by approximation and satisfy a Löwenheim-Skolem condition.
Cuntz semigroups of ideals and quotients and a generalized Kasparov Stabilization Theorem
设A是C* 代数，I是A的闭双边理想.利用Cuntz半群的Hilbert C ~*-模图研究了I，A和A/I的Cuntz半群之间的关系。得到了A的Cuntz半群中两个元素的一个关系，刻画了它们在A/I的Cuntz半群中相等的条件.作为推论，我们证明了Cuntz半群函子是正合的。用Hilbert模的同构代替Hilbert模的Cuntz等价关系，得到了Kasparov稳定性定理的一个推广.
We provide an abstract characterization for the Cuntz semigroup of unital commutative AI-algebras, as well as a characterization for abstract Cuntz semigroups of the form $\text{Lsc} (X,\overline{\mathbb{N}})$ for some $T_1$-space $X$. In our investigations, we also uncover new properties that the Cuntz semigroup of all AI-algebras satisfies.
We give a local characterization for the Cuntz semigroup of AI-algebras building upon Shen's characterization of dimension groups. Using this result, we provide an abstract characterization for the Cuntz semigroup of AI-algebras.
This paper argues that the unitary Cuntz semigroup, introduced in [10] and termed Cu$_1$, contains crucial information regarding the classification of non-simple C$^*$-algebras. We exhibit two (non-simple) C$^*$-algebras that agree on their Cuntz semigroups, termed Cu, and their K$_1$-groups and yet disagree at level of their unitary Cuntz semigroups. In the process, we establish that the unitary Cuntz semigroup contains rigorously more information about non-simple C$^*$-algebras than Cu and K$_1$ alone.
We introduce a notion of covering dimension for Cuntz semigroups of C*-algebras. This dimension is always bounded by the nuclear dimension of the C*-algebra, and for subhomogeneous C*-algebras both dimensions agree. Cuntz semigroups of Z-stable C*-algebras have dimension at most one. Further, the Cuntz semigroup of a simple, Z-stable C*-algebra is zero-dimensional if and only if the C*-algebra has real rank zero or is stably projectionless.
In this paper, we show that for unital, separable $C^*$-algebras of stable rank one and real rank zero, the unitary Cuntz semigroup functor and the functor ${\rm K}_*$ are naturallly equivalent. Then we introduce a refinement of the unitary Cuntz semigroup, say the total Cuntz semigroup, which is a new invariant for separable $C^*$-algebras of stable rank one, is a well-defined continuous functor from the category of $C^*$-algebras of stable rank one to the category ${\rm\underline{ Cu}}$. We prove that this new functor and the functor ${\rm \underline{K}}$ are naturallly equivalent for unital…
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)\s…
We study topological aspects of the category of abstract Cuntz semigroups, termed Cu. We provide a suitable setting in which we are able to uniformly control how to approach an element of a Cu-semigroup by a rapidly increasing sequence. This approximation induces a semimetric on the set of Cu-morphisms, generalizing Cu-metrics that had been constructed in the past for some particular cases. Further, we develop an approximate intertwining theory for the category Cu. Finally, we give several applications such as the classification of unitary elements of any unital AF-algebra by means of the func…
Let $A$ be a separable simple exact ${\cal Z}$-stable $C^*$-algebra. We show that the unitay group of ${\tilde A}$ has the cancellation property. If $A$ has continuous scale, the Cuntz semigroup of $\tilde A$ has the strict comparison property and a weak cancellation property. Let $C$ be a 1-dimensional non-commutative CW complex with $K_1(C)=\{0\}.$ Suppose that $λ: {\rm Cu}^\sim(C)\to {\rm Cu}^\sim(A)$ is a morphism in Cuntz semigroups which is strictly positive. Then there exists a sequence of homomorphisms $ϕ_n: C\to A$ such that $\lim_{n\to\infty}{\rm Cu}^\sim(ϕ_n)=λ.$ This result leads t…
We introduce a new invariant for C*-algebras of stable rank one that merges the Cuntz semigroup information together with the K$_1$-group information. This semigroup, termed the 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 Cu$_1$-semigroup is a well-defined continuous functor from the category of C*-algebras of stable rank one to a suitable codomain category tha…
We give a detailed introduction to the theory of Cuntz semigroups for C*-algebras. Beginning with the most basic definitions and technical lemmas, we present several results of historical importance, such as Cuntz's theorem on the existence of quasitraces, Rørdam's proof that $\mathcal{Z}$-stability implies strict comparison, and Toms' example of a non $\mathcal{Z}$-stable simple, nuclear C*-algebra. We also give the reader an extensive overview of the state of the art and the modern approach to the theory, including the recent results for C*-algebras of stable rank one (for example, the Black…
