We show that for $\mathrm {C}^*$-algebras with the global Glimm property, the rank of every operator can be realized as the rank of a soft operator, that is, an element whose hereditary sub-$\mathrm {C}^*$-algebra has no nonzero, unital quotients. This implies that the radius of comparison of such a $\mathrm {C}^*$-algebra is determined by the soft part of its Cuntz semigroup.

Under a mild additional assumption, we show that every Cuntz class dominates a (unique) largest soft Cuntz class. This defines a retract from the Cuntz semigroup onto its soft part, and it follows that the covering dimensions of these semigroups differ by at most $1$.

We elaborate on the construction of the Evans chain complex for higher-rank graph $C^*$-algebras. Specifically, we introduce a block matrix presentation of the differential maps. These block matrices are then used to identify a wide family of higher-rank graph $C^*$-algebras with trivial K-theory. Additionally, in the specialised case where the higher-rank graph consists of one vertex, we are able to use the Künneth theorem to explicitly compute the homology groups of the Evans chain complex.

In this article, we introduce and study the notion of Goldie dimension for C*-algebras. We prove that a C*-algebra A has Goldie dimension n if and only if the dimension of the center of its local multiplier algebra is n. In this case, A has finite-dimensional center and its primitive spectrum is extremally disconnected. If moreover, A is extending, we show that it decomposes into a direct sum of n prime C*-algebras. In particular, every stably finite, exact C*-algebra with Goldie dimension, that has the projection property and a strictly full element, admits a full projection and a non-zero densely defined lower semi-continuous trace. Finally we show that certain C*-algebras with Goldie dimension (not necessarily simple, separable or nuclear) are classifiable by the Elliott invariant.

We show that continuous group homomorphisms between unitary groups of unital C*-algebras induce maps between spaces of continuous real-valued affine functions on the trace simplices. Under certain $K$-theoretic regularity conditions, these maps can be seen to commute with the pairing between $K_0$ and traces. If the homomorphism is contractive and sends the unit circle to the unit circle, the map between spaces of continuous real-valued affine functions can further be shown to be unital and positive (up to a minus sign).

We prove that crossed products of fiberwise essentially minimal zero-dimensional dynamical systems, a class that includes systems in which all orbit closures are minimal, have isomorphic K-theory if and only if the dynamical systems are strong orbit equivalent. Under the additional assumption that the dynamical systems have no periodic points, this gives a classification theorem including isomorphism of the associated crossed product $C^*$-algebras as well. We additionally explore the K-theory of such crossed products and the Bratteli diagrams associated to the dynamical systems.

We construct two new classes of topological dynamical systems; one is a factor of a one-sided shift of finite type while the second is a factor of the two-sided shift. The data are a finite graph which presents the shift of finite type, a second finite directed graph and a pair of embeddings of it into the first, satisfying certain conditions. The factor is then obtained from a simple idea based on binary expansion of real numbers. In both cases, we construct natural metrics on the factors and, in the second case, this makes the system a Smale space, in the sense of Ruelle. We compute various algebraic invariants for these systems, including the homology for Smale space developed by the author and the K-theory of various $C^{*}$-algebras associated to them, in terms of the pair of original graphs.

We show that the properties of being rationally K-stable passes from the fibres of a continuous $C(X)$-algebra to the ambient algebra, under the assumption that the underlying space X is compact, metrizable, and of finite covering dimension. As an application, we show that a crossed product C*-algebra is (rationally) K-stable provided the underlying C*-algebra is (rationally) K-stable, and the action has finite Rokhlin dimension with commuting towers.

Scarparo has constructed counterexamples to Matui’s HK-conjecture. These counterexamples and other known counterexamples are essentially principal but not principal. In the present paper, a counterexample to the HK-conjecture that is principal is given. Like Scarparo’s original counterexample, our counterexample is the transformation groupoid associated to a particular odometer. However, the relevant group is the fundamental group of a flat manifold (and hence is torsion-free) and the associated odometer action is free. The examples discussed here do satisfy the rational version of the HK-conjecture.

In this paper, let A be an infinite-dimensional stably finite unital simple separable $\mathrm {C^*}$ -algebra. Let $B\subset A$ be a centrally large subalgebra in A such that B has uniform property $\Gamma $ . Then we prove that A has uniform property $\Gamma $ . Let $\Omega $ be a class of stably finite unital $\mathrm {C^*}$ -algebras such that for any $B\in \Omega $ , B has uniform property $\Gamma $ . Then we show that A has uniform property $\Gamma $ for any simple unital $\mathrm {C^*}$ -algebra $A\in \rm {TA}\Omega $ .

We obtain a characterization of the unital C*-algebras with the property that every element is a limit of products of positive elements, thereby answering a question of Murphy and Phillips.

Let G be a finite group, let A be an infinite-dimensional stably finite simple unital C*-algebra, and let $\alpha \colon G \to {\operatorname {Aut}} (A)$ be an action of G on A which has the weak tracial Rokhlin property. Let $A^{\alpha }$ be the fixed point algebra. Then the radius of comparison satisfies ${\operatorname {rc}} (A^{\alpha }) \leq {\operatorname {rc}} (A)$ and ${\operatorname {rc}} ( C^* (G, A, \alpha ) ) \leq ({1}/{{{\operatorname{card}}} (G))} \cdot {\operatorname {rc}} (A)$ . The inclusion of $A^{\alpha }$ in A induces an isomorphism from the purely positive part of the Cuntz semigroup ${\operatorname {Cu}} (A^{\alpha })$ to the fixed points of the purely positive part of ${\operatorname {Cu}} (A)$ , and the purely positive part of ${\operatorname {Cu}} ( C^* (G, A, \alpha ) )$ is isomorphic to this semigroup. We construct an example in which $G \,{=}\, {\mathbb {Z}} / 2 {\mathbb {Z}}$ , A is a simple unital AH algebra, $\alpha $ has the Rokhlin property, ${\operatorname {rc}} (A)> 0$ , ${\operatorname {rc}} (A^{\alpha }) = {\operatorname {rc}} (A)$ , and ${\operatorname {rc}} ({C^* (G, A, \alpha)} ) = ({1}/{2}) {\operatorname {rc}} (A)$ .

Since their inception in the 1930s by von Neumann, operator algebras have been used to shed light on many mathematical theories. Classification results for self-adjoint and non-self-adjoint operator algebras manifest this approach, but a clear connection between the two has been sought since their emergence in the late 1960s. We connect these seemingly separate types of results by uncovering a hierarchy of classification for non-self-adjoint operator algebras and $C^{*}$-algebras with additional $C^{*}$-algebraic structure. Our approach naturally applies to algebras arising from $C^{*}$-correspondences to resolve self-adjoint and non-self-adjoint isomorphism problems in the literature. We apply our strategy to completely elucidate this newly found hierarchy for operator algebras arising from directed graphs.

We construct a Baum–Connes assembly map localised at the unit element of a discrete group $\Gamma$. This morphism, called $\mu _\tau$, is defined in $KK$-theory with coefficients in $\mathbb {R}$ by means of the action of the idempotent $[\tau ]\in KK_{\mathbin {{\mathbb {R}}}}^\Gamma (\mathbb {C},\mathbb {C})$ canonically associated to the group trace of $\Gamma$. We show that the corresponding $\tau$-Baum–Connes conjecture is weaker than the classical version, but still implies the strong Novikov conjecture. The right-hand side of $\mu _\tau$ is functorial with respect to the group $\Gamma$.

We introduce a dimension group for a self-similar map as the $\mathrm {K}_0$ -group of the core of the C*-algebra associated with the self-similar map together with the canonical endomorphism. The key step for the computation is an explicit description of the core as the inductive limit using their matrix representations over the coefficient algebra, which can be described explicitly by the singularity structure of branched points. We compute that the dimension group for the tent map is isomorphic to the countably generated free abelian group ${\mathbb Z}^{\infty }\cong {\mathbb Z}[t]$ together with the unilateral shift, i.e. the multiplication map by t as an abstract group. Thus the canonical endomorphisms on the $\mathrm {K}_0$ -groups are not automorphisms in general. This is a different point compared with dimension groups for topological Markov shifts. We can count the singularity structure in the dimension groups.

We study compact group actions with finite Rokhlin dimension, particularly in relation to crossed products. For example, we characterize the duals of such actions, generalizing previous partial results for the Rokhlin property. As an application, we determine the ideal structure of their crossed products. Under the assumption of so-called commuting towers, we show that taking crossed products by such actions preserves a number of relevant classes of $C^{\ast }$-algebras, including: $D$-absorbing $C^{\ast }$-algebras, where $D$ is a strongly self-absorbing $C^{\ast }$-algebra; stable $C^{\ast }$-algebras; $C^{\ast }$-algebras with finite nuclear dimension (or decomposition rank); $C^{\ast }$-algebras with finite stable rank (or real rank); and $C^{\ast }$-algebras whose $K$-theory is either trivial, rational, or $n$-divisible for $n\in \mathbb{N}$. The combination of nuclearity and the universal coefficient theorem (UCT) is also shown to be preserved by these actions. Some of these results are new even in the well-studied case of the Rokhlin property. Additionally, and under some technical assumptions, we show that finite Rokhlin dimension with commuting towers implies the (weak) tracial Rokhlin property. At the core of our arguments is a certain local approximation of the crossed product by a continuous $C(X)$-algebra with fibers that are stably isomorphic to the underlying algebra. The space $X$ is computed in some cases of interest, and we use its description to construct a $\mathbb{Z}_{2}$-action on a unital AF-algebra and on a unital Kirchberg algebra satisfying the UCT, whose Rokhlin dimensions with and without commuting towers are finite but do not agree.

We show that Matui’s HK conjecture holds for groupoids of unstable equivalence relations and their corresponding $C^{\ast }$-algebras on one-dimensional solenoids.

A crossed product functor is said to be injective if it takes injective morphisms to injective morphisms. In this paper we show that every locally compact group $G$ admits a maximal injective crossed product $A\mapsto A\rtimes _{\text{inj}}G$. Moreover, we give an explicit construction of this functor that depends only on the maximal crossed product and the existence of $G$-injective $C^{\ast }$-algebras; this is a sort of ‘dual’ result to the construction of the minimal exact crossed product functor, the latter having been studied for its relationship to the Baum–Connes conjecture. It turns out that $\rtimes _{\text{inj}}$ has interesting connections to exactness, the local lifting property, amenable traces, and the weak expectation property.

Wieler has shown that every irreducible Smale space with totally disconnected stable sets is a solenoid (i.e., obtained via a stationary inverse limit construction). Using her construction, we show that the associated stable $C^{\ast }$-algebra is the stationary inductive limit of a $C^{\ast }$-stable Fell algebra that has a compact spectrum and trivial Dixmier–Douady invariant. This result applies in particular to Williams solenoids along with other examples. Beyond the structural implications of this inductive limit, one can use this result to, in principle, compute the $K$-theory of the stable $C^{\ast }$-algebra. A specific one-dimensional Smale space (the $aab/ab$-solenoid) is considered as an illustrative running example throughout.

We compute the homology groups of transformation groupoids associated with odometers and show that certain $(\mathbb{Z}\rtimes \mathbb{Z}_{2})$-odometers give rise to counterexamples to the HK conjecture, which relates the homology of an essentially principal, minimal, ample groupoid $G$ with the K-theory of $C_{r}^{\ast }(G)$. We also show that transformation groupoids of odometers satisfy the AH conjecture.

The semigroups of unital extensions of separable C*-algebras come in two flavours: a strong and a weak version. By the unital Ext-groups, we mean the groups of invertible elements in these semigroups. We use the unital Ext-groups to obtain K-theoretic classification of both unital and non-unital extensions of C*-algebras, and in particular we obtain a complete K-theoretic classification of full extensions of UCT Kirchberg algebras by stable AF algebras.