Cuntz Semigroup Research Articles

The modern theory of Cuntz semigroups of $C^{\ast}$-algebras

We give a detailed introduction to the theory of Cuntz semigroups for C^{\ast} -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^{\ast} -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^{\ast} -algebras of stable rank one (for example, the Blackadar–Handelman conjecture and the realization of ranks), as well as the abstract study of the Cuntz category Cu .

Fraïssé theory for Cuntz semigroups

We develop a theory of Cauchy sequences and intertwinings for morphisms of Cuntz semigroups, which generalizes all past approaches to study metric-like properties of the invariant. Further, the techniques presented here can be applied to all known refinements of the Cuntz semigroup, including those that may be used in new classification results.As a particular application, we introduce a Fraïssé theory for abstract Cuntz semigroups akin to the theory of Fraïssé categories developed by Kubiś. We also show that any (Cuntz) Fraïssé category has a unique Fraïssé limit which is both universal and homogeneous. Several examples of such categories and their Fraïssé limits are given throughout the paper.

Total Cuntz semigroup, extension, and Elliott Conjecture with real rank zero

AbstractIn this paper, we exhibit two unital, separable, nuclear ‐algebras of stable rank one and real rank zero with the same ordered scaled total K‐theory, but they are not isomorphic with each other, which forms a counterexample to the Elliott Classification Conjecture for real rank zero setting. Thus, we introduce an additional normal condition and give a classification result in terms of the total K‐theory. For the general setting, with a new invariant, the total Cuntz semigroup [2], we classify a large class of ‐algebras obtained from extensions. The total Cuntz semigroup, which distinguishes the algebras of our counterexample, could possibly classify all the ‐algebras of stable rank one and real rank zero.

Tracial oscillation zero and stable rank one

Abstract Let A be a separable (not necessarily unital) simple $C^*$ -algebra with strict comparison. We show that if A has tracial approximate oscillation zero, then A has stable rank one and the canonical map $\Gamma $ from the Cuntz semigroup of A to the corresponding lower-semicontinuous affine function space is surjective. The converse also holds. As a by-product, we find that a separable simple $C^*$ -algebra which has almost stable rank one must have stable rank one, provided it has strict comparison and the canonical map $\Gamma $ is surjective.

Finite approximation properties of C*-modules III

Abstract We introduce and study a notion of module nuclear dimension for a C * \mathrm{C}^{*} -algebra A which is a C * \mathrm{C}^{*} -module over another C * \mathrm{C}^{*} -algebra 𝔄 {\mathfrak{A}} with compatible actions. We show that the module nuclear dimension of A is zero if A is 𝔄 {\mathfrak{A}} -NF. The converse is shown to hold when 𝔄 {\mathfrak{A}} is a C ⁢ ( X ) {C(X)} -algebra with simple fibers, with X compact and totally disconnected. We also introduce a notion of module decomposition rank, and show that when 𝔄 {\mathfrak{A}} is unital and simple, if the module decomposition rank of A is finite then A is 𝔄 {\mathfrak{A}} -QD. We study the set 𝒯 𝔄 ⁢ ( A ) {\mathcal{T}_{\mathfrak{A}}(A)} of 𝔄 {\mathfrak{A}} -valued module traces on A and relate the Cuntz semigroup of A with lower semicontinuous affine functions on the set 𝒯 𝔄 ⁢ ( A ) {\mathcal{T}_{\mathfrak{A}}(A)} . Along the way, we also prove a module Choi–Effros lifting theorem. We give estimates of the module nuclear dimension for a class of examples.

Cones of traces arising from AF $C^{*}$-algebras

We characterize the topological non-cancellative cones that can be expressed as projective limits of finite powers of [0,\infty] . For metrizable cones, these are also the cones of lower semicontinuous extended-valued traces on approximately finite-dimensional (AF) C^{*} -algebras. Our main result may be regarded as a generalization of the fact that any Choquet simplex is a projective limit of finite-dimensional simplices. To obtain our main result, we first establish a duality between certain non-cancellative topological cones and Cuntz semigroups with real multiplication. This duality extends the duality between compact convex sets and complete order unit vector spaces to a non-cancellative setting.

Malcolmson semigroups

Inspired by the construction of the Cuntz semigroup for a C⁎-algebra, we introduce the matrix Malcolmson semigroup and the finitely presented module Malcolmson semigroup for a unital ring. These two semigroups are shown to have isomorphic Grothendieck group in general and be isomorphic for von Neumann regular rings. For unital C⁎-algebras, it is shown that the matrix Malcolmson semigroup has a natural surjective order-preserving homomorphism to the Cuntz semigroup, every dimension function is a Sylvester matrix rank function, and there exist Sylvester matrix rank functions which are not dimension functions.

The Global Glimm Property

It is known that a C ∗ C^* -algebra with the Global Glimm Property is nowhere scattered (it has no elementary ideal-quotients), and the Global Glimm Problem asks if the converse holds. We provide a new approach to this long-standing problem by showing that a C ∗ C^* -algebra has the Global Glimm Property if and only if it is nowhere scattered and its Cuntz semigroup is ideal-filtered (the Cuntz classes generating a given ideal are downward directed) and has property (V) (a weak form of being sup-semilattice ordered). We show that ideal-filteredness and property (V) are automatic for C ∗ C^* -algebras that have stable rank one or real rank zero, thereby recovering the solutions to the Global Glimm Problem in these cases. We also use our approach to solve the Global Glimm Problem for new classes of C ∗ C^* -algebras.

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 K⁎ are natural 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 Cu_. We prove that this new functor and the functor K_ are naturally equivalent for unital, separable, K-pure C⁎-algebras. Therefore, the total Cuntz semigroup is a complete invariant for a large class of C⁎-algebras of real rank zero.

The unitary Cuntz semigroup on the classification of non-simple C⁎-algebras

This paper argues that the unitary Cuntz semigroup, introduced in [10] and termed Cu1, contains crucial information regarding the classification of non-simple C⁎-algebras. We exhibit two non-simple C⁎-algebras that agree on their K1-groups and their Cuntz semigroups (termed Cu) 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 K1 alone.

The Cuntz semigroup of unital commutative AI-algebras

AbstractWe 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 $\operatorname {\mathrm {Lsc}} (X,\overline {\mathbb {N}})$ for some $T_1$ -space X. In our investigations, we also uncover new properties that the Cuntz semigroup of all AI-algebras satisfies.

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 functor Cu.

Stable elements and property (S)

We study the relation (and differences) between stability and Property (S) in the simple and stably finite framework. This leads us to characterize stable elements in terms of their support, and study these concepts from different sides: hereditary subalgebras, projections in the multiplier algebra and order properties in the Cuntz semigroup. We use these approaches to show both that cancellation at infinity on the Cuntz semigroup just holds when its Cuntz equivalence is given by isomorphism at the level of Hilbert right-modules, and that different notions such as Regularity, ω-comparison, Corona Factorization Property, property R, etc. are equivalent under mild assumptions.

Unitary groups and augmented Cuntz semigroups of separable simple 𝒵-Stable C∗-algebras

Let [Formula: see text] be a separable simple exact [Formula: see text]-stable [Formula: see text]-algebra. We show that the unitary group of [Formula: see text] has the cancellation property. If [Formula: see text] has continuous scale then the Cuntz semigroup of [Formula: see text] has strict comparison property and a weak cancellation property. Let [Formula: see text] be a 1-dimensional noncommutative CW complex with [Formula: see text] Suppose that [Formula: see text] is a morphism in the augmented Cuntz semigroups which is strictly positive. Then there exists a sequence of homomorphisms [Formula: see text] such that [Formula: see text] This result leads to the proof that every separable amenable simple [Formula: see text]-algebra[Formula: see text]in the UCT class has rationally generalized tracial rank at most one.

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

The uncovering of new structure on the Cuntz semigroup of a C*-algebra of stable rank one leads to several applications: we answer affirmatively, for the class of stable rank-one C*-algebras, a conjecture by Blackadar and Handelman on dimension functions, the global Glimm halving problem, and the problem of realizing functions on the cone of 2-quasitraces as ranks of Cuntz semigroup elements. We also gain new insights into the comparability properties of positive elements in C*-algebras of stable rank one.

Distance between unitary orbits in C∗-algebras with stable rank one and real rank zero

Let A be a C∗-algebra with stable rank one and real rank zero. In this paper, it is shown that the usual distance dU defined on the approximate unitary equivalence classes (or unitary orbits) of the positive elements in A is equal to the distance dW defined on morphisms from Cuntz semigroup of C0(0,1] to the Cuntz semigrout of A.

Covering dimension of Cuntz semigroups II

We show that the dimension of the Cuntz semigroup of a [Formula: see text]-algebra is determined by the dimensions of the Cuntz semigroups of its separable sub-[Formula: see text]-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 [Formula: see text]-algebra by sub-[Formula: see text]-algebras. We show that many properties for Cuntz semigroups are preserved by approximation and satisfy a Löwenheim–Skolem condition.

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.

A local characterization for the Cuntz semigroup of AI-algebras

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.

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 K1-group information. This semigroup, termed the Cu1-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 Cu1-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 Cu∼. Furthermore, we compute the Cu1-semigroup of some specific classes of C⁎-algebras. Finally, in the course of our investigation, we show that we can recover functorially Cu, K1 and K⁎:=K0⊕K1 from Cu1.