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 $1$.

Let $\mathfrak{C}$ be the smallest class of countable discrete groups with the following properties: (i) $\mathfrak{C}$ contains the trivial group, (ii) $\mathfrak{C}$ is closed under isomorphisms, countable increasing unions and extensions by $\mathbb{Z}$. Note that $\mathfrak{C}$ contains all countable discrete torsion-free abelian groups and poly-$\mathbb{Z}$ groups. Also, $\mathfrak{C}$ is a subclass of the class of countable discrete torsion-free elementary amenable groups. In this article, we show that if $\Gamma\in \mathfrak{C}$, then all strongly outer actions of Γ on the Razak–Jacelon algebra $\mathcal{W}$ are cocycle conjugate to each other. This can be regarded as an analogous result of Szabó’s result for strongly self-absorbing C$^*$-algebras.

Given any unital, finite, classifiable $\mathrm{C}^*$-algebra A with real rank zero and any compact simplex bundle with the fibre at zero being homeomorphic to the space of tracial states on A, we show that there exists a flow on A realizing this simplex. Moreover, we show that given any unital $\mathrm{UCT}$ Kirchberg algebra A and any proper simplex bundle with empty fibre at zero, there exists a flow on A realizing this simplex.

When $\mathcal {D}$ is strongly self-absorbing, we say an inclusion $B \subseteq A$ of C*-algebras is $\mathcal {D}$-stable if it is isomorphic to the inclusion $B \otimes \mathcal {D} \subseteq A \otimes \mathcal {D}$. We give ultrapower characterizations and show that if a unital inclusion is $\mathcal {D}$-stable, then $\mathcal {D}$-stability can be exhibited for countably many intermediate C*-algebras concurrently. We show that such unital embeddings between unital $\mathcal {D}$-stable C*-algebras are point-norm dense in the set of all unital embeddings, and that every unital embedding between $\mathcal {D}$-stable C*-algebras is approximately unitarily equivalent to a $\mathcal {D}$-stable embedding. Examples are provided.

We discuss a strategy for classifying anomalous actions through model action absorption. We use this to upgrade existing classification results for Rokhlin actions of finite groups on C$^*$-algebras, with further assuming a UHF-absorption condition, to a classification of anomalous actions on these C$^*$-algebras.

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.

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.

Consider a minimal-free topological dynamical system $(X, \mathbb Z^d)$. It is shown that the radius of comparison of the crossed product C*-algebra $\mathrm {C}(X) \rtimes \mathbb Z^d$ is at most half the mean topological dimension of $(X, \mathbb Z^d)$. As a consequence, the C*-algebra $\mathrm {C}(X) \rtimes \mathbb Z^d$ is classified by the Elliott invariant if the mean dimension of $(X, \mathbb Z^d)$ is zero.

The action on the trace space induced by a generic automorphism of a suitable finite classifiable ${\mathrm {C}^*}$ -algebra is shown to be chaotic and weakly mixing. Model ${\mathrm {C}^*}$ -algebras are constructed to observe the central limit theorem and other statistical features of strongly chaotic tracial actions. Genericity of finite Rokhlin dimension is used to describe $KK$ -contractible stably projectionless ${\mathrm {C}^*}$ -algebras as crossed products.

What is the probability that a random UHF algebra is of infinite type? What is the probability that a random simple AI algebra has at most k extremal traces? What is the expected value of the radius of comparison of a random Villadsen-type AH algebra? What is the probability that such an algebra is $\mathcal{Z}$-stable? What is the probability that a random Cuntz–Krieger algebra is purely infinite and simple, and what can be said about the distribution of its K-theory? By constructing $\mathrm{C}^*$-algebras associated with suitable random (walks on) graphs, we provide context in which these are meaningful questions with computable answers.

We investigate the notion of tracial ${\mathcal {Z}}$-stability beyond unital $\mathrm {C}^*$-algebras, and we prove that this notion is equivalent to ${\mathcal {Z}}$-stability in the class of separable simple nuclear $\mathrm {C}^*$-algebras.

We investigate quantum lens spaces, $C(L_q^{2n+1}(r;\underline {m}))$, introduced by Brzeziński and Szymański as graph $C^*$-algebras. We give a new description of $C(L_q^{2n+1}(r;\underline {m}))$ as graph $C^*$-algebras amending an error in the original paper by Brzeziński and Szymański. Furthermore, for $n\leq 3$, we give a number-theoretic invariant, when all but one weight are coprime to the order of the acting group r. This builds upon the work of Eilers, Restorff, Ruiz, and Sørensen.

It is shown that the colored isomorphism class of a unital, simple, $\mathcal {Z}$-stable, separable amenable C$^*$-algebra satisfying the universal coefficient theorem is determined by its tracial simplex.

We construct two types of unital separable simple $C^*$ -algebras: $A_z^{C_1}$ and $A_z^{C_2}$ , one exact but not amenable, the other nonexact. Both have the same Elliott invariant as the Jiang–Su algebra – namely, $A_z^{C_i}$ has a unique tracial state,

$$ \begin{align*} \left(K_0\left(A_z^{C_i}\right), K_0\left(A_z^{C_i}\right)_+, \left[1_{A_z^{C_i}} \right]\right)=(\mathbb{Z}, \mathbb{Z}_+,1), \end{align*} $$

and $K_{1}\left (A_z^{C_i}\right )=\{0\}$ ( $i=1,2$ ). We show that $A_z^{C_i}$ ( $i=1,2$ ) is essentially tracially in the class of separable ${\mathscr Z}$ -stable $C^*$ -algebras of nuclear dimension $1$ . $A_z^{C_i}$ has stable rank one, strict comparison for positive elements and no $2$ -quasitrace other than the unique tracial state. We also produce models of unital separable simple nonexact (exact but not nuclear) $C^*$ -algebras which are essentially tracially in the class of simple separable nuclear ${\mathscr Z}$ -stable $C^*$ -algebras, and the models exhaust all possible weakly unperforated Elliott invariants. We also discuss some basic properties of essential tracial approximation.

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 characterize topological conjugacy classes of one-sided topological Markov shifts in terms of the associated Cuntz–Krieger algebras and their gauge actions with potentials.

The Toms–Winter conjecture is verified for those separable, unital, nuclear, infinite-dimensional real C*-algebras for which the complexification has a tracial state space with compact extreme boundary of finite covering dimension.

We revisit the notion of tracial approximation for unital simple $C^*$ -algebras. We show that a unital simple separable infinite dimensional $C^*$ -algebra A is asymptotically tracially in the class of $C^*$ -algebras with finite nuclear dimension if and only if A is asymptotically tracially in the class of nuclear $\mathcal {Z}$ -stable $C^*$ -algebras.

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.