We introduce and study the weak Glimm property for $\mathrm{C}^{*}$-algebras, and also a property we shall call (HS$_0$). We show that the properties of being nowhere scattered and residual (HS$_0$) are equivalent for any $\mathrm{C}^{*}$-algebra. Also, for a $\mathrm{C}^{*}$-algebra with the weak Glimm property, the properties of being purely infinite and weakly purely infinite are equivalent. It follows that for a $\mathrm{C}^{*}$-algebra with the weak Glimm property such that the absolute value of every nonzero, square-zero, element is properly infinite, the properties of being (weakly, locally) purely infinite, nowhere scattered, residual (HS$_0$), residual (HS$_{\text {t}}$), and residual (HI) are all equivalent, and are equivalent to the global Glimm property. This gives a partial affirmative answer to the global Glimm problem, as well as certain open questions raised by Kirchberg and Rørdam.

We describe two kinds of regular invariant measures on the boundary path space $\partial E$ of a second countable topological graph E, which allows us to describe all extremal tracial weights on $C^{*}(E)$ which are not gauge-invariant. Using this description, we prove that all tracial weights on the C$^{*}$-algebra $C^{*}(E)$ of a second countable topological graph E are gauge-invariant when E is free. This in particular implies that all tracial weights on $C^{*}(E)$ are gauge-invariant when $C^{*}(E)$ is simple and separable.

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.

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 the Hilbert bimodule associated with a compact topological graph can be recovered from the $C^*$-algebraic triple consisting of the Toeplitz algebra of the graph, its gauge action and the commutative subalgebra of functions on the vertex space of the graph. We discuss connections with work of Davidson–Katsoulis and of Davidson–Roydor on local conjugacy of topological graphs and isomorphism of their tensor algebras. In particular, we give a direct proof that a compact topological graph can be recovered up to local conjugacy from its Hilbert bimodule, and present an example of nonisomorphic locally conjugate compact topological graphs with isomorphic Hilbert bimodules. We also give an elementary proof that for compact topological graphs with totally disconnected vertex space the notions of local conjugacy, Hilbert bimodule isomorphism, isomorphism of $C^*$-algebraic triples, and isomorphism all coincide.

We investigate almost minimal actions of abelian groups and their crossed products. As an application, given multiplicatively independent integers p and q, we show that Furstenberg’s $\times p,\times q$ conjecture holds if and only if the canonical trace is the only faithful extreme tracial state on the $C^*$-algebra of the group $\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2$. We also compute the primitive ideal space and K-theory of $C^*(\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2)$.

We show that all values in the interval $[0,{\pi }/{2}]$ can be attained as interior angles between intermediate subalgebras (as introduced by Bakshi and the first named author [‘Lattice of intermediate subalgebras’, J. Lond. Math. Soc. (2) 104(2) (2021), 2082–2127]) of a certain inclusion of simple unital $C^*$-algebras. We also calculate the interior angles between intermediate crossed product subalgebras of any inclusion of crossed product algebras corresponding to any action of a countable discrete group and its subgroups on a unital $C^*$-algebra.

We initiate the study of computable presentations of real and complex C*-algebras under the program of effective metric structure theory. With the group situation as a model, we develop corresponding notions of recursive presentations and word problems for C*-algebras, and show some analogous results hold in this setting. Famously, every finitely generated group with a computable presentation is computably categorical, but we provide a counterexample in the case of C*-algebras. On the other hand, we show every finite-dimensional C*-algebra is computably categorical.

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.

Let $C_c^{*}(\mathbb{N}^{2})$ be the universal $C^{*}$-algebra generated by a semigroup of isometries $\{v_{(m,n)}\,:\, m,n \in \mathbb{N}\}$ whose range projections commute. We analyse the structure of KMS states on $C_{c}^{*}(\mathbb{N}^2)$ for the time evolution determined by a homomorphism $c\,:\,\mathbb{Z}^{2} \to \mathbb{R}$. In contrast to the reduced version $C_{red}^{*}(\mathbb{N}^{2})$, we show that the set of KMS states on $C_{c}^{*}(\mathbb{N}^{2})$ has a rich structure. In particular, we exhibit uncountably many extremal KMS states of type I, II and III.

For every minimal one-sided shift space X over a finite alphabet, left special elements are those points in X having at least two preimages under the shift operation. In this paper, we show that the Cuntz–Pimsner $C^*$-algebra $\mathcal {O}_X$ has nuclear dimension $1$ when X is minimal and the number of left special elements in X is finite. This is done by describing concretely the cover of X, which also recovers an exact sequence, discovered before by Carlsen and Eilers.

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 apply a method inspired by Popa's intertwining-by-bimodules technique to investigate inner conjugacy of MASAs in graph $C^*$-algebras. First, we give a new proof of non-inner conjugacy of the diagonal MASA ${\mathcal {D}}_E$ to its non-trivial image under a quasi-free automorphism, where $E$ is a finite transitive graph. Changing graphs representing the algebras, this result applies to some non quasi-free automorphisms as well. Then, we exhibit a large class of MASAs in the Cuntz algebra ${\mathcal {O}}_n$ that are not inner conjugate to the diagonal ${\mathcal {D}}_n$.

We prove a double commutant theorem for separable subalgebras of a wide class of corona C*-algebras, largely resolving a problem posed by Pedersen in 1988. Double commutant theorems originated with von Neumann, whose seminal result evolved into an entire field now called von Neumann algebra theory. Voiculescu later proved a C*-algebraic double commutant theorem for subalgebras of the Calkin algebra. We prove a similar result for subalgebras of a much more general class of so-called corona C*-algebras.

For amenable discrete groupoids $\mathcal {G}$ and row-finite directed graphs E, let $(\mathcal {G},E)$ be a self-similar groupoid and let $C^*(\mathcal {G}, E)$ be the associated $C^*$-algebra. We introduce a weaker faithfulness condition than those in the existing literature that still guarantees that $C^*(\mathcal {G})$ embeds in $C^*(\mathcal {G}, E)$. Under this faithfulness condition, we prove a gauge-invariant uniqueness theorem.

We present here a multiplicative version of the classical Kowalski–Słodkowski theorem, which identifies the characters among the collection of all functionals on a complex and unital Banach algebra A. In particular, we show that, if A is a $C^\star $-algebra, and if $\phi :A\to \mathbb C $ is a continuous function satisfying $ \phi (x)\phi (y) \in \sigma (xy) $ for all $x,y\in A$ (where $\sigma $ denotes the spectrum), then either $\phi $ is a character of A or $-\phi $ is a character of A.

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

For a given inverse semigroup action on a topological space, one can associate an étale groupoid. We prove that there exists a correspondence between the certain subsemigroups and the open wide subgroupoids in case that the action is strongly tight. Combining with the recent result of Brown et al., we obtain a correspondence between the certain subsemigroups of an inverse semigroup and the Cartan intermediate subalgebras of a groupoid C*-algebra.

In Kiukas, Lahti, and Ylinen (2006, Journal of Mathematical Physics 47, 072104), the authors asked the following general question. When is a positive operator measure projection valued? A version of this question formulated in terms of operator moments was posed in Pietrzycki and Stochel (2021, Journal of Functional Analysis 280, 109001). Let T be a self-adjoint operator, and let F be a Borel semispectral measure on the real line with compact support. For which positive integers $p< q$ do the equalities $T^k =\int _{\mathbb {R}} x^k F(\mathrm {d\hspace {.1ex}} x)$, $k=p, q$, imply that F is a spectral measure? In the present paper, we completely solve the second problem. The answer is affirmative if $p$ is odd and $q$ is even, and negative otherwise. The case $(p,q)=(1,2)$ closely related to intrinsic noise operator was solved by several authors including Kruszyński and de Muynck, as well as Kiukas, Lahti, and Ylinen. The counterpart of the second problem concerning the multiplicativity of unital positive linear maps on $C^*$-algebras is also provided.