In this paper, we initiate the study of higher rank Baumslag–Solitar (BS) semigroups and their related C*-algebras. We focus on two rather interesting classes—one is related to products of odometers and the other is related to Furstenberg’s $\times p, \times q$ conjecture. For the former class, whose C*-algebras are studied in [32], we here characterize the factoriality of the associated von Neumann algebras and further determine their types; for the latter, we obtain their canonical Cartan subalgebras. In the rank 1 case, we study a more general setting that encompasses (single-vertex) generalized BS semigroups. One of our main tools in this paper is from self-similar higher rank graphs and their C*-algebras.

We extend the definition of the X-Rokhlin property to countable discrete groups and prove some permanence properties. If the action of a countable discrete group on X is free and minimal and the action of this group on the separable simple $C^*$-algebra has the X-Rokhlin property, then the reduced crossed product is simple.

We give a unified overview of the study of the effects of additional set theoretic axioms on quotient structures. Our focus is on rigidity, measured in terms of existence (or rather non-existence) of suitably non-trivial automorphisms of the quotients in question. A textbook example for the study of this topic is the Boolean algebra $\mathcal {P}({\mathbb N})/\operatorname {\mathrm {Fin}}$, whose behavior is the template around which this survey revolves: Forcing axioms imply that all of its automorphisms are trivial, in the sense that they are induced by almost permutations of ${\mathbb N}$, while under the Continuum Hypothesis this rigidity fails and $\mathcal {P}({\mathbb N})/\operatorname {\mathrm {Fin}}$ admits uncountably many non-trivial automorphisms. We consider far-reaching generalisations of this phenomenon and present a wide variety of situations where analogous patterns persist, focusing mainly (but not exclusively) on the categories of Boolean algebras, Čech–Stone remainders, and $\mathrm {C}^{*}$-algebras. We survey the state of the art and the future prospects of this field, discussing the major open problems and outlining the main ideas of the proofs whenever possible.

We provide a characterization of equivariant Fock covariant injective representations for product systems. We show that this characterization coincides with Nica covariance for compactly aligned product systems over right least common multiple semigroups of Kwaśniewski and Larsen and with the Toeplitz representations of a discrete monoid of Laca and Sehnem. By combining with the framework established by Katsoulis and Ramsey, we resolve the reduced Hao–Ng isomorphism problem for generalized gauge actions by discrete groups.

Let $\Omega $ be a compact subset of $\mathbb {C}$ and let A be a unital simple, separable $C^*$-algebra with stable rank one, real rank zero, and strict comparison. We show that, given a Cu-morphism ${\alpha :\mathrm { Cu}(C(\Omega ))\to \mathrm {Cu}(A)}$ with , there exists a homomorphism $\phi : C(\Omega )\to A$ such that $\mathrm {Cu}(\phi )=\alpha $. Moreover, if $K_1(A)$ is trivial, then $\phi $ is unique up to approximate unitary equivalence. We also give classification results for maps from a large class of $C^*$-algebras to A in terms of the Cuntz semigroup.

We write arbitrary separable nuclear $\mathrm {C}^*$-algebras as limits of inductive systems of finite-dimensional $\mathrm {C}^*$-algebras with completely positive connecting maps. The characteristic feature of such ${\mathrm {CPC}^*}$-systems is that the maps become more and more orthogonality preserving. This condition makes it possible to equip the limit, a priori only an operator space, with a multiplication turning it into a $\mathrm {C}^*$-algebra. Our concept generalizes the NF systems of Blackadar and Kirchberg beyond the quasidiagonal case.

The concept of stability has proved very useful in the field of Banach space geometry. In this note, we introduce and study a corresponding concept in the setting of Banach algebras, which we call multiplicative stability. As we shall prove, various interesting examples of Banach algebras are multiplicatively unstable, and hence unstable in the model-theoretic sense. The examples include Fourier algebras over noncompact amenable groups, $C^*$-algebras and the measure algebra of an infinite compact group.

We prove that a C$^*$-algebra A has uniform property $\Gamma $ if the set of extremal tracial states, $\partial _e T(A)$, is a non-empty compact space of finite covering dimension and for each $\tau \in \partial _e T(A)$, the von Neumann algebra $\pi _\tau (A)"$ arising from the GNS representation has property $\Gamma $.

Inspired by work of Szymik and Wahl on the homology of Higman–Thompson groups, we establish a general connection between ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces, based on the construction of small permutative categories of compact open bisections. This allows us to analyse homological invariants of topological full groups in terms of homology for ample groupoids.

Applications include complete rational computations, general vanishing and acyclicity results for group homology of topological full groups as well as a proof of Matui’s AH-conjecture for all minimal, ample groupoids with comparison.

In this article, it is shown that the lattice of C$^*$-covers of an operator algebra does not contain enough information to distinguish operator algebras up to completely isometric isomorphism. In addition, four natural equivalences of the lattice of C$^*$-covers are developed and proven to be distinct. The lattice of C$^*$-covers of direct sums and tensor products are studied. Along the way key examples are found of operator algebras, each of which generates exactly n C$^*$-algebras up to $*$-isomorphism, and a simple operator algebra that is not similar to a C$^*$-algebra.

Given a full right-Hilbert $\mathrm {C}^{*}$-module $\mathbf {X}$ over a $\mathrm {C}^{*}$-algebra A, the set $\mathbb {K}_{A}(\mathbf {X})$ of A-compact operators on $\mathbf {X}$ is the (up to isomorphism) unique $\mathrm {C}^{*}$-algebra that is strongly Morita equivalent to the coefficient algebra A via $\mathbf {X}$. As a bimodule, $\mathbb {K}_{A}(\mathbf {X})$ can also be thought of as the balanced tensor product $\mathbf {X}\otimes _{A} \mathbf {X}^{\mathrm {op}}$, and so the latter naturally becomes a $\mathrm {C}^{*}$-algebra. We generalize both of these facts to the world of Fell bundles over groupoids: Suppose $\mathscr {B}$ is a Fell bundle over a groupoid $\mathcal {H}$ and $\mathscr {M}$ is an upper semi-continuous Banach bundle over a principal $\mathcal {H}$-space X. If $\mathscr {M}$ carries a right-action of $\mathscr {B}$ and a sufficiently nice $\mathscr {B}$-valued inner product, then its imprimitivity Fell bundle $\mathbb {K}_{\mathscr {B}}(\mathscr {M})=\mathscr {M}\otimes _{\mathscr {B}} \mathscr {M}^{\mathrm {op}}$ is a Fell bundle over the imprimitivity groupoid of X, and it is the unique Fell bundle that is equivalent to $\mathscr {B}$ via $\mathscr {M}$. We show that $\mathbb {K}_{\mathscr {B}}(\mathscr {M})$ generalizes the “higher order” compact operators of Abadie–Ferraro in the case of saturated bundles over groups, and that the theorem recovers results such as Kumjian’s Stabilization trick.

Let R be a ring and let $n\ge 2$. We discuss the question of whether every element in the matrix ring $M_n(R)$ is a product of (additive) commutators $[x,y]=xy-yx$, for $x,y\in M_n(R)$. An example showing that this does not always hold, even when R is commutative, is provided. If, however, R has Bass stable rank one, then under various additional conditions every element in $M_n(R)$ is a product of three commutators. Further, if R is a division ring with infinite center, then every element in $M_n(R)$ is a product of two commutators. If R is a field and $a\in M_n(R)$, then every element in $M_n(R)$ is a sum of elements of the form $[a,x][a,y]$ with $x,y\in M_n(R)$ if and only if the degree of the minimal polynomial of a is greater than $2$.

For an integral domain R satisfying certain conditions, we characterise the primitive ideal space and its Jacobson topology for the semigroup crossed product $C^*(R_+) \rtimes R^\times $. We illustrate the result by the example $R=\mathbb {Z}[\sqrt {-3}]$.

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 $1$.

By employing the external Kasparov product, in [18], Hawkins, Skalski, White, and Zacharias constructed spectral triples on crossed product C$^\ast $-algebras by equicontinuous actions of discrete groups. They further raised the question for whether their construction turns the respective crossed product into a compact quantum metric space in the sense of Rieffel. By introducing the concept of groups separated with respect to a given length function, we give an affirmative answer in the case of virtually Abelian groups equipped with certain orbit metric length functions. We further complement our results with a discussion of natural examples such as generalized Bunce-Deddens algebras and higher-dimensional noncommutative tori.

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.

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.