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.

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.

Fundamental group of a manifold gives a deep effect on its underlying smooth structure. In this paper we introduce a new variant of the Donaldson invariant in Yang–Mills gauge theory from twisting by the Picard group of a 4-manifold in the case when the fundamental group is free abelian. We then generalise it to the general case of fundamental groups by use of the framework of non commutative geometry. We also verify that our invariant distinguishes smooth structures between some homeomorphic 4-manifolds.

Given a unital inductive limit of C*-algebras for which each C*-algebra of the inductive sequence comes equipped with a Rieffel compact quantum metric, we produce sufficient conditions to build a compact quantum metric on the inductive limit from the quantum metrics on the inductive sequence by utilizing the completeness of the dual Gromov–Hausdorff propinquity of Latrémolière on compact quantum metric spaces. This allows us to place new quantum metrics on all unital approximately finite-dimensional (AF) algebras that extend our previous work with Latrémolière on unital AF algebras with faithful tracial state. As a consequence, we produce a continuous image of the entire Fell topology on the ideal space of any unital AF algebra in the dual Gromov–Hausdorff propinquity topology.

We provide a systematic study of a non-commutative extension of the classical Anzai skew-product for the cartesian product of two copies of the unit circle to the non-commutative 2-tori. In particular, some relevant ergodic properties are proved for these quantum dynamical systems, extending the corresponding ones enjoyed by the classical Anzai skew-product. As an application, for a uniquely ergodic Anzai skew-product $\unicode[STIX]{x1D6F7}$ on the non-commutative $2$-torus $\mathbb{A}_{\unicode[STIX]{x1D6FC}}$, $\unicode[STIX]{x1D6FC}\in \mathbb{R}$, we investigate the pointwise limit, $\lim _{n\rightarrow +\infty }(1/n)\sum _{k=0}^{n-1}\unicode[STIX]{x1D706}^{-k}\unicode[STIX]{x1D6F7}^{k}(x)$, for $x\in \mathbb{A}_{\unicode[STIX]{x1D6FC}}$ and $\unicode[STIX]{x1D706}$ a point in the unit circle, and show that there are examples for which the limit does not exist, even in the weak topology.

The modular Gromov–Hausdorff propinquity is a distance on classes of modules endowed with quantum metric information, in the form of a metric form of a connection and a left Hilbert module structure. This paper proves that the family of Heisenberg modules over quantum two tori, when endowed with their canonical connections, form a family of metrized quantum vector bundles, as a first step in proving that Heisenberg modules form a continuous family for the modular Gromov–Hausdorff propinquity.

Fix a von Neumann algebra ${\mathcal{N}}$ equipped with a suitable trace $\unicode[STIX]{x1D70F}$. For a path of self-adjoint Breuer–Fredholm operators, the spectral flow measures the net amount of spectrum that moves from negative to non-negative. We consider specifically the case of paths of bounded perturbations of a fixed unbounded self-adjoint Breuer–Fredholm operator affiliated with ${\mathcal{N}}$. If the unbounded operator is $p$-summable (that is, its resolvents are contained in the ideal $L^{p}$), then it is possible to obtain an integral formula that calculates spectral flow. This integral formula was first proved by Carey and Phillips, building on earlier approaches of Phillips. Their proof was based on first obtaining a formula for the larger class of $\unicode[STIX]{x1D703}$-summable operators, and then using Laplace transforms to obtain a $p$-summable formula. In this paper, we present a direct proof of the $p$-summable formula that is both shorter and simpler than theirs.

Let $\mathbb{L}$ be a length function on a group $G$, and let ${{M}_{\mathbb{L}}}$ denote the operator of pointwise multiplication by $\mathbb{L}$ on ${{\ell }^{2}}\left( G \right)$. Following Connes, ${{M}_{\mathbb{L}}}$ can be used as a “Dirac” operator for the reduced group ${{C}^{*}}$-algebra $C_{r}^{*}\left( G \right)$. It defines a Lipschitz seminorm on $C_{r}^{*}\left( G \right)$, which defines a metric on the state space of $C_{r}^{*}\left( G \right)$. We show that for any length function satisfying a strong form of polynomial growth on a discrete group, the topology from this metric coincides with the weak-$*$ topology (a key property for the definition of a “compact quantum metric space”). In particular, this holds for all word-length functions on finitely generated nilpotent-by-finite groups.

Let $A\,\in \,{{M}_{n}}\left( \mathbb{R} \right)$ be an invertible matrix. Consider the semi-direct product ${{\mathbb{R}}^{n}}\rtimes \,\mathbb{Z}$ where the action of $\mathbb{Z}$ on ${{\mathbb{R}}^{n}}$ is induced by the left multiplication by $A$. Let $\left( \alpha ,\,\tau \right)$ be a strongly continuous action of ${{\mathbb{R}}^{n}}\rtimes \,\mathbb{Z}$ on a ${{C}^{*}}$-algebra $B$ where $\alpha$ is a strongly continuous action of ${{\mathbb{R}}^{n}}$ and $\tau$ is an automorphism. The map $\tau$ induces a map $\widetilde{\tau }\,\text{on}\,\text{B}\,{{\rtimes }_{\alpha }}\,{{\mathbb{R}}^{n}}$. We show that, at the $K$-theory level, $\tau$ commutes with the Connes–Thom map if $\det \left( A \right)\,>\,0$ and anticommutes if $\det \left( A \right)\,>\,0$. As an application, we recompute the $K$-groups of the Cuntz–Li algebra associated with an integer dilation matrix.

We show that the functor from curved differential graded algebras to differential graded categories, defined by the second author, sends Cartesian diagrams to homotopy Cartesian diagrams, under certain reasonable hypotheses. This is an extension to the arena of dg-categories of a construction of projective modules due to Milnor. As an example, we show that the functor satisfies descent for certain partitions of a complex manifold.

The excision theorem of Cuntz and Quillen established the existence of a six term exact sequence in the bivariant periodic cyclic cohomology HP*(–,–) associated with an arbitrary algebra extension 0 → S → P → Q → 0. This remarkable result enabled far reaching developments in the purely algebraic periodic cyclic cohomology. It also provided a new formalism that led to the creation of new versions of this theory for topological and bornological algebras. In this article we outline some of the developments that resulted from this breakthrough.

In this note we show that the crucial orientation condition forcommutative geometries fails for the natural commutative spectral triple of an orbifold M/G.

The classical Fourier–Mukai duality establishes an equivalence of categories between the derived categories of sheaves on dual complex tori. In this article we show that this equivalence extends to an equivalence between two dual objects. Both of these are generalized deformations of the complex tori. In one case, a complex torus is deformed formally in a non-commutative direction specified by a holomorphic Poisson structure. In the other, the dual complex torus is deformed in a $B$-field direction to a formal gerbe. We show that these two deformations are Fourier–Mukai equivalent.

We study the $\text{K}$-homology of the rotation algebras ${{A}_{\theta }}$ using the six-term cyclic sequence for the $\text{K}$-homology of a crossed product by $Z$. In the case that $\theta $ is irrational, we use Pimsner and Voiculescu's work on $\text{AF}$-embeddings of the ${{A}_{\theta }}$ to search for the missing generator of the even $\text{K}$-homology.