The Cuntz semigroup as an invariant for C*-algebras

The Cuntz semigroup of continuous functions into certain simple C*-algebras

This paper contains computations of the Cuntz semigroup of separable C*-algebras of the form C_0(X,A), where A is a unital, simple, Z-stable ASH algebra. The computations describe the Cuntz semigroup in terms of Murray-von Neumann semigroups of C(K,A) for compact subsets K of X. In particular, the computation shows that the Elliott invariant is functorially equivalent to the invariant given by the Cuntz semigroup of C(T,A). These results are a contribution towards the goal of using the Cuntz semigroup in the classification of well-behaved non-simple C*-algebras.

A revised augmented Cuntz semigroup

We revise the construction of the augmented Cuntz semigroup functor used by the first author to classify inductive limits of 1-dimensional noncommutative CW complexes. The original construction has good functorial properties when restricted to the class of C*-algebras of stable rank one. The construction proposed here has good properties for all C*-algebras: We show that the augmented Cuntz semigroup is a stable, continuous, split exact functor, from the category of C*-algebras to the category of Cu-semigroups.

Cuntz-Li relations, Inverse semigroups and Groupoids

In this paper we show that the universal C*-algebra satisfying the Cuntz-Li relations is generated by an inverse semigroup of partial isometries. We apply Exel's theory of tight representations to this inverse semigroup. We identify the universal C*-algebra as the C*-algebra of the tight groupoid associated to the inverse semigroup.

The Cuntz semigroup of the tensor product C*-algebras

We calculate the Cuntz semigroup of the tensor product A with A. We restrict our attention to C*-algebras A which are unital, simple, nuclear, stably finite, have stable rank one, absorbs the Jiang-Su algebra tensorially and satisfy the UCT.

Covering dimension of Cuntz semigroups

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.

单位图 C*-代数的 Cuntz 半群

The Cuntz semigroup of a unital graph C*-algebra

本文结合多种技术计算单位图 C*-代数的 Cuntz 半群，研究其性质，并证明在 Cuntz 半群层面无碍于核维数至多为一的猜想。

Cuntz semigroups of ultraproduct C*-algebras

We prove that the category of abstract Cuntz semigroups is bicomplete. As a consequence, the category admits products and ultraproducts. We further show that the scaled Cuntz semigroup of the (ultra)product of a family of C*-algebras agrees with the (ultra)product of the scaled Cuntz semigroups of the involved C*-algebras. As applications of our results, we compute the non-stable K-Theory of general (ultra)products of C*-algebras and we characterize when ultraproducts are simple. We also give criteria that determine order properties of these objects, such as almost unperforation.

A total Cuntz semigroup for $C^*$-algebras of stable rank one

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…

The Cuntz semigroup of some spaces of dimension at most two

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…

通过动力学Cuntz半群对逆半群交叉积的二分法

A dichotomy for inverse-semigroup crossed products via dynamical Cuntz semigroups

研究逆半群作用下C* - 代数本质交叉积的稳定有限性与纯无限性，通过诱导作用发展动力学Cuntz半群，证明其允许或不允许非平凡态决定交叉积性质，推广了相关结果。

Stability in the Cuntz semigroup of a commutative C*-algebra

We prove stability theorems in the Cuntz semigroup of a commutative C*-algebra which are analogues of classical stability theorems for topological vector bundles over compact Hausdorff spaces. Several applications to simple unital AH algebras of slow dimension growth are then given: such algebras have strict comparison of positive elements; their Cuntz semigroups are recovered functorially from the Elliott invariant; the lower-semicontinuous dimension functions are dense in the space of all dimension functions, and the latter is a Choquet simplex.

Cuntz's $ax+b$-semigroup C*-algebra over $\mathbb{N}$ and product system C*-algebras

We investigate $C^*$-algebras associated with row-finite topological higher-rank graphs with no source, which are based on product system $C^*$-algebras. We prove the Cuntz-Krieger uniqueness theorem, and provide the condition of simplicity and purely infiniteness of our algebras. We give examples of non-discrete topological higher-rank graphs whose $C^*$-algebras contain Cuntz's $ax+b$-semigroup $C^*$-algebra over $\mathbb{N}$.

Three Applications of the Cuntz Semigroup

Building on work of Elliott and coworkers, we present three applications of the Cuntz semigroup: (i) for many simple C$^*$-algebras, the Thomsen semigroup is recovered functorially from the Elliott invariant, and this yields a new proof of Elliott's classification theorem for simple, unital AI algebras; (ii) for the algebras in (i), classification of their Hilbert modules is similar to the von Neumann algebra context; (iii) for the algebras in (i), approximate unitary equivalence of self-adjoint operators is characterised in terms of the Elliott invariant.

The universal Boolean inverse semigroup presented by the abstract Cuntz-Krieger relations

This paper is a contribution to the theory of what might be termed $0$-dimensional non-commutative spaces. We prove that associated with each inverse semigroup $S$ is a Boolean inverse semigroup presented by the abstract versions of the Cuntz-Krieger relations. We call this Boolean inverse semigroup the Exel completion of $S$ and show that it arises from Exel's tight groupoid under non-commutative Stone duality.

The cone of functionals on the Cuntz semigroup

The functionals on an ordered semigroup S in the category Cu--a category to which the Cuntz semigroup of a C*-algebra naturally belongs--are investigated. After appending a new axiom to the category Cu, it is shown that the "realification" S_R of S has the same functionals as S and, moreover, is recovered functorially from the cone of functionals of S. Furthermore, if S has a weak Riesz decomposition property, then S_R has refinement and interpolation properties which imply that the cone of functionals on S is a complete distributive lattice. These results apply to the Cuntz semigroup of a C*-…

Unitary Cuntz semigroups of ideals and quotients

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…

Uniformly Based Cuntz semigroups and approximate intertwinings

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…

The Cuntz Semigroup and Comparison of Open Projections

We show that a number of naturally occurring comparison relations on positive elements in a C*-algebra are equivalent to natural comparison properties of their corresponding open projections in the bidual of the C*-algebra. In particular we show that Cuntz comparison of positive elements corresponds to a comparison relation on open projections, that we call Cuntz comparison, and which is defined in terms of-and is weaker than-a comparison notion defined by Peligrad and Zsidó. The latter corresponds to a well-known comparison relation on positive elements defined by Blackadar. We show that Murr…

Unitary groups and augmented Cuntz semigroups of separable simple Z-stable C*-algebras

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…

The Corona Factorization property, Stability, and the Cuntz semigroup of a C*-algebra

The Corona Factorization Property, originally invented to study extensions of C*-algebras, conveys essential information about the intrinsic structure of the C*-algebras. We show that the Corona Factorization Property of a σ-unital C*-algebra is completely captured by its Cuntz semigroup (of equivalence classes of positive elements in the stabilization of A). The corresponding condition in the Cuntz semigroup is a very weak comparability property termed the Corona Factorization Property for semigroups. Using this result one can for example show that all unital C*-algebras with finite decomposi…

The Cuntz semigroup and stability of close C*-algebras

We prove that separable C*-algebras which are completely close in a natural uniform sense have isomorphic Cuntz semigroups, continuing a line of research developed by Kadison - Kastler, Christensen, and Khoshkam. This result has several applications: we are able to prove that the property of stability is preserved by close C*-algebras provided that one algebra has stable rank one; close C*-algebras must have affinely homeomorphic spaces of lower-semicontinuous quasitraces; strict comparison is preserved by sufficient closeness of C*-algebras. We also examine C*-algebras which have a positive a…

稳定秩一的 C*-代数及其 Cuntz 半群

C*-algebras of stable rank one and their Cuntz semigroups

本文通过揭示稳定秩一 C*-代数的 Cuntz 半群新结构，解决了维数函数猜想、Global Glimm Halving 及秩实现问题，并深化了正元素比较性质的理解。

A unitary Cuntz semigroup for C*-algebras of stable rank one

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…

动力Cuntz半群与Cuntz半群的无理想商

The dynamical Cuntz semigroup and ideal-free quotients of Cuntz semigroups

本文发展了W-和Cu-半群的一般商理论，引入正规对概念定义动力Cuntz半群，并在C*-代数框架下利用其泛性探究交叉积C*-代数的Cuntz半群结构。

抽象双变量 Cuntz 半群 II

Abstract Bivariant Cuntz Semigroups II

本文延续抽象 Cuntz 半群研究，构造闭对称幺半范畴中的复合与外部张量积，并刻画非交换 Cu-半环，得到实心 Cu-半环的新等价条件。

Comparison Properties of the Cuntz semigroup and applications to C*-algebras

We study comparison properties in the category Cu aiming to lift results to the C*-algebraic setting. We introduce a new comparison property and relate it to both the CFP and $ω$-comparison. We show differences of all properties by providing examples, which suggest that the corona factorization property for C*-algebras might allow for both finite and infinite projections. In addition, we show that Rørdam's simple, nuclear C*-algebra with a finite and an infinite projection does not have the CFP.

Recovering the Elliott invariant from the Cuntz semigroup

Let $A$ be a simple, separable C$^*$-algebra of stable rank one. We prove that the Cuntz semigroup of $\CC(\T,A)$ is determined by its Murray-von Neumann semigroup of projections and a certain semigroup of lower semicontinuous functions (with values in the Cuntz semigroup of $A$). This result has two consequences. First, specializing to the case that $A$ is simple, finite, separable and $\mathcal Z$-stable, this yields a description of the Cuntz semigroup of $\CC(\T,A)$ in terms of the Elliott invariant of $A$. Second, suitably interpreted, it shows that the Elliott functor and the functor def…

全 Cuntz 半群、扩张与实秩零的 Elliott 猜想

Total Cuntz semigroup, Extension and Elliott Conjecture with Real rank zero

本文构造了实秩零情形下 Elliott 分类猜想的反例，并引入全 Cuntz 半群作为新不变量，对由扩张得到的一大类 C*-代数给出分类结果。

A Bivariant Theory for the Cuntz Semigroup

We introduce a bivariant version of the Cuntz semigroup as equivalence classes of order zero maps generalizing the ordinary Cuntz semigroup. The theory has many properties formally analogous to KK-theory including a composition product. We establish basic properties, like additivity, stability and continuity, and study categorical aspects in the setting of local C*-algebras. We determine the bivariant Cuntz semigroup for numerous examples such as when the second algebra is a Kirchberg algebra, and Cuntz homology for compact Hausdorff spaces which provides a complete invariant. Moreover, we est…

Inverse semigroups and the Cuntz-Li algebras

In this paper, we apply the theory of inverse semigroups to the $C^{*}$-algebra $U[\mathbb{Z}]$ considered in \cite{Cuntz}. We show that the $C^{*}$-algebra $U[\mathbb{Z}]$ is generated by an inverse semigroup of partial isometries. We explicity identify the groupoid $\mathcal{G}_{tight}$ associated to the inverse semigroup and show that $\mathcal{G}_{tight}$ is exactly the same groupoid obtained in \cite{Cuntz-Li}.

MF traces and the Cuntz semigroup

A trace $τ$ on a separable C*-algebra $A$ is called matricial field (MF) if there is a trace-preserving morphism from $A$ to $Q_ω$, where $Q_ω$ denotes the norm ultrapower of the universal UHF-algebra $Q$. In general, the trace $τ$ induces a state on the Cuntz semigroup $Cu(A)$. We show there is always a state-preserving morphism from $Cu(A)$ to $Cu(Q_ω)$. As an application, if $A$ is an AI-algebra and $F$ is a free group acting on $A$, then every trace on the reduced crossed product $A \rtimes F$ is MF. This further implies the same result when $A$ is an AH-algebra with the ideal property suc…

Completions of monoids with applications to the Cuntz semigroup

We provide an abstract categorical framework that relates the Cuntz semigroups of the C$^*$-algebras $A$ and $A\otimes \mathcal{K}$. This is done through a certain completion of ordered monoids by adding suprema of countable ascending sequences. Our construction is rather explicit and we show it is functorial and unique up to isomorphism. This approach is used in some applications to compute the stabilized Cuntz semigroup of certain C$^*$-algebras.

Pullbacks, $C(X)$-algebras, and their Cuntz semigroup

In this paper we analyse the structure of the Cuntz semigroup of certain $C(X)$-algebras, for compact spaces of low dimension, that have no $\mathrm{K}_1$-obstruction in their fibres in a strong sense. The techniques developed yield computations of the Cuntz semigroup of some surjective pullbacks of C$^*$-algebras. As a consequence, this allows us to give a complete description, in terms of semigroup valued lower semicontinuous functions, of the Cuntz semigroup of $C(X,A)$, where $A$ is a not necessarily simple C$^*$-algebra of stable rank one and vanishing $\mathrm{K}_1$ for each closed, two …

Cuntz semigroups of compact-type Hopf C*-algebras

The classical Cuntz semigroup has an important role in the study of C*-algebras, being one of the main invariants used to classify recalcitrant C*-algebras up to isomorphism. We consider C*-algebras that have Hopf algebra structure, and find additional structure in their Cuntz semigroups, thus generalizing the equivariant Cuntz semigroup. We develop various aspects of the theory of such semigroups, and in particular, we give general results allowing classification results of the Elliott classification program to be extended to classification results for C*-algebraic quantum groups.

The Cuntz semigroup, a Riesz type interpolation property, comparison and the ideal property

We define a Riesz type interpolation property for the Cuntz semigroup of a $C^*$-algebra and prove it is satisfied by the Cuntz semigroup of every $C^*$-algebra with the ideal property. Related to this, we obtain two characterizations of the ideal property in terms of the Cuntz semigroup of the $C^*$-algebra. Some additional characterizations are proved in the special case of the stable, purely infinite $C^*$-algebras, and two of them are expressed in language of the Cuntz semigroup. We introduce a notion of comparison of positive elements for every unital $C^*$-algebra that has (normalized) q…

An Equivariant Theory for the Bivariant Cuntz Semigroup

We provide an equivariant extension of the bivariant Cuntz semigroup introduced in previous work for the case of compact group actions over C*-algebras. Its functoriality properties are explored and some well-known classification results are retrieved. Connections with crossed products are investigated, and a concrete presentation of equivariant Cuntz homology is provided. The theory that is here developed can be used to define the equivariant Cuntz semigroup. We show that the object thus obtained coincides with the recently proposed one by Gardella and Santiago, and we complement their work b…

The Cuntz semigroup and the radius of comparison of the crossed product by a finite group

Let G be a finite group, let A be an infinite-dimensional stably finite simple unital C*-algebra, and let α\colon G \to Aut (A) be an action of G on A which has the weak tracial Rokhlin property. Let A^α be the fixed point algebra. Then the radius of comparison satisfies rc (A^α) \leq rc (A) and rc ( C* (G, A, α) ) \leq ( 1 / card (G) ) rc (A). The inclusion of A^α in A induces an isomorphism from the purely positive part of the Cuntz semigroup Cu (A^α) to the fixed points of the purely positive part of Cu (A), and the purely positive part of Cu ( C* (G, A, α) ) is isomorphic to this semigroup…

Open Projections and Suprema in the Cuntz Semigroup

We provide a new and concise proof of the existence of suprema in the Cuntz semigroup using the open projection picture of the Cuntz semigroup initiated by Ortega, Rordam and Thiel. Our argument is based on the observation that the supremum of a countable set of open projections in the bidual of a C*-algebra A is again open and corresponds to the generated hereditary sub-C*-algebra of A.

Higher-rank graph C*-algebras: an inverse semigroup and groupoid approach

We provide inverse semigroup and groupoid models for the Toeplitz and Cuntz-Krieger algebras of finitely aligned higher-rank graphs. Using these models, we prove a uniqueness theorem for the Cuntz-Krieger algebra.

Cuntz半群的Fraïssé理论

Fraïssé Theory for Cuntz semigroups

本文为抽象Cuntz半群引入Fraïssé理论，证明任何(Cuntz)Fraïssé范畴存在唯一且万有、齐次的Fraïssé极限，并给出实例及在范畴Cu中发展Cauchy序列与交织理论。

The equivariant Cuntz semigroup

We introduce an equivariant version of the Cuntz semigroup, that takes an action of a compact group into account. The equivariant Cuntz semigroup is naturally a semimodule over the representation semiring of the given group. Moreover, this semimodule satisfies a number of additional structural properties. We show that the equivariant Cuntz semigroup, as a functor, is continuous and stable. Moreover, cocycle conjugate actions have isomorphic associated equivariant Cuntz semigroups. One of our main results is an analog of Julg's theorem: the equivariant Cuntz semigroup is canonically isomorphic …

The Cuntz semigroup of continuous fields

In this paper we describe the Cuntz semigroup of continuous fields of C$^*$-algebras over one dimensional spaces whose fibers have stable rank one and trivial $K_1$ for each closed, two-sided ideal. This is done in terms of the semigroup of global sections on a certain topological space built out of the Cuntz semigroups of the fibers of the continuous field. When the fibers have furthermore real rank zero, and taking into account the action of the space, our description yields that the Cuntz semigroup is a classifying invariant if and only if so is the sheaf induced by the Murray-von Neumann s…

The Cuntz semigroup and domain theory

Domain theory has its origins in Mathematics and Theoretical Computer Science. Mathematically it combines order and topology. Its central concepts have their origin in the idea of approximating ideal objects by their relatively finite or, more generally, relatively compact parts. The development of domain theory in recent years was mainly motivated by question in denotational semantics and the theory of computation. But since 2008, domain theoretical notions and methods are used in the theory of C*-algebras in connection with the Cuntz semigroup. This paper is largely expository. It presents t…

Topological aperiodicity for product systems over semigroups of Ore type

We prove a version of uniqueness theorem for Cuntz-Pimsner algebras of discrete product systems over semigroups of Ore type. To this end, we introduce Doplicher-Roberts picture of Cuntz-Pimsner algebras, and the semigroup dual to a product system of 'regular' C*-correspondences. Under a certain aperiodicity condition on the latter, we obtain the uniqueness theorem and a simplicity criterion for the algebras in question. These results generalize the corresponding ones for crossed products by discrete groups, due to Archbold and Spielberg, and for Exel's crossed products, due to Exel and Vershik…

The K-theory of some reduced inverse semigroup C*-algebras

We use a recent result by Cuntz, Echterhoff and Li about the K-theory of certain reduced C*-crossed products to describe the K-theory of C*_r(S) when S is an inverse semigroup satisfying certain requirements. A result of Milan and Steinberg allows us to show that C*_r(S) is Morita equivalent to a crossed product of the type handled by Cuntz, Echterhoff and Li. We apply our result to graph inverse semigroups and the inverse semigroups of one-dimensional tilings.

The modern theory of Cuntz semigroups of C*-algebras

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…

环的 Cuntz 半群

The Cuntz semigroup of a ring

本文为任意环定义由可数生成投射模构造的 Cuntz 半群 $\mathrm{S}(R)$，给出其等价描述与扩展不变量 $\mathrm{SCu}(R)$，并在多类环中计算且关联 C*-代数情形。

稳定秩一的环的Cuntz半群

The Cuntz semigroup of rings with stable rank one

本文研究稳定秩一环的Cuntz半群，证明Cuntz次等价等价于纯子模同构，Cuntz等价即同构，并建立有限生成投射模的序消去性质。

On the structure of the Cuntz semigroup in (possibly) nonunital C*-algebras

We examine the ranks of operators in semi-finite C*-algebras as measured by their densely defined lower semicontinuous traces. We first prove that a unital simple C*-algebra whose extreme tracial boundary is nonempty and finite contains positive operators of every possible rank, independent of the property of strict comparison. We then turn to nonunital simple algebras and establish criteria that imply that the Cuntz semigroup is recovered functorially from the Murray-von Neumann semigroup and the space of densely defined lower semicontinuous traces. Finally, we prove that these criteria are s…

Crossed products by abelian semigroups via transfer operators

We propose a generalisation of Exel's crossed product by a single endomorphism and a transfer operator to the case of actions of abelian semigroups of endomorphisms and associated transfer operators. The motivating example for our definition yields new crossed products, not obviously covered by familiar theory. Our technical machinery builds on Fowler's theory of Toeplitz and Cuntz-Pimsner algebras of discrete product systems of Hilbert bimodules, which we need to expand to cover a natural notion of relative Cuntz-Pimsner algebras of product systems.

Tensor products and regularity properties of Cuntz semigroups

The Cuntz semigroup of a C*-algebra is an important invariant in the structure and classification theory of C*-algebras. It captures more information than K-theory but is often more delicate to handle. We systematically study the lattice and category theoretic aspects of Cuntz semigroups. Given a C*-algebra $A$, its (concrete) Cuntz semigroup $Cu(A)$ is an object in the category $Cu$ of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu. To clarify the distinction between concrete and abstract Cuntz semigroups, we will call the latter $Cu$-semigroups. We establish the e…

扩张C*-代数的某些Cuntz半群性质

Certain Cuntz semigroup properties of extension C*-algebras

本文研究扩张C*-代数的Cuntz半群维数，证明维数上界满足加法不等式，并继承比较与可除性性质，给出纯性等价的新证明。

The Cuntz semigroup, the Elliott conjecture, and dimension functions on C*-algebras

We prove that the Cuntz semigroup is recovered functorially from the Elliott invariant for a large class of C*-algebras. In particular, our results apply to the largest class of simple C*-algebras for which K-theoretic classification can be hoped for. This work has three significant consequences. First, it provides new conceptual insight into Elliott's classification program, proving that the usual form of the Elliott conjecture is equivalent, among Z-stable algebras, to a conjecture which is in general substantially weaker and for which there are no known counterexamples. Second and third, it…

Regularity for stably projectionless, simple C*-algebras

This paper explores the following regularity properties and their relationships for simple, not-necessarily-unital C*-algebras: (i) Jiang-Su stability, (ii) Unperforation in the Cuntz semigroup, and (iii) slow dimension growth (applying only in the case that the C*-algebra is approximately subhomogeneous). An example is given of a simple, separable, nuclear, stably projectionless C*-algebra whose Cuntz semigroup is not almost unperforated. This example is in fact approximately subhomogeneous. It is also shown that, in contrast to this example, when an approximately subhomogeneous simple C*-alg…