We investigate the Lorentzian analogues of Riemannian Bianchi–Cartan–Vranceanu spaces. We provide their general description and emphasize their role in the classification of three-dimensional homogeneous Lorentzian manifolds with a four-dimensional isometry group. We then illustrate their geometric properties (with particular regard to curvature, Killing vector fields and their description as Lorentzian Lie groups) and we study several relevant classes of surfaces (parallel, totally umbilical, minimal, constant mean curvature) in these homogeneous Lorentzian three-manifolds.

Jones [‘Two subfactors and the algebraic decomposition of bimodules over $II_1$ factors’, Acta Math. Vietnam 33(3) (2008), 209–218] proposed the study of ‘two subfactors’ of a $II_1$ factor as a quantization of two closed subspaces in a Hilbert space. Motivated by this, we initiate a systematic study of a special class of two subfactors, namely a pair of spin model subfactors. We characterize which pairs of distinct complex Hadamard matrices in $M_n(\mathbb {C})$ give rise to distinct spin model subfactors. Then, a detailed investigation is carried out for $n=2$, where the spin model subfactors correspond to $\mathbb {Z}_2$-actions on the hyperfinite type $II_1$ factor R. We observe that the intersection of the pair of spin model subfactors in this case is a nonirreducible vertex model subfactor and we characterize it as a diagonal subfactor. A few key invariants for the pair of spin model subfactors are computed to understand their relative positions.

We introduce positive correspondences as right $C^*$-modules with left actions given by completely positive maps. Positive correspondences form a semi-category that contains the $C^*$-correspondence (Enchilada) category as a ‘retract’. Kasparov’s KSGNS construction provides a semi-functor from this semi-category onto the $C^*$-correspondence category. The need for left actions by completely positive maps appears naturally when we consider morphisms between Cuntz–Pimsner algebras, and we describe classes of examples arising from projections on $C^*$-correspondences and Fock spaces, as well as examples from conjugation by bi-Hilbertian bimodules of finite index.

For a connected Lie group G and an automorphism T of G, we consider the action of T on Sub$_G$, the compact space of closed subgroups of G endowed with the Chabauty topology. We study the action of T on Sub$^p_G$, the closure in Sub$_G$ of the set of closed one-parameter subgroups of G. We relate the distality of the T-action on Sub$^p_G$ with that of the T-action on G and characterise the same in terms of compactness of the closed subgroup generated by T in Aut$(G)$ when T acts distally on the maximal central torus and G is not a vector group. We extend these results to the action of a subgroup of Aut$(G)$ and equate the distal action of any closed subgroup ${\mathcal H}$ on Sub$^p_G$ with that of every element in ${\mathcal H}$. Moreover, we show that a connected Lie group G acts distally on Sub$^p_G$ by conjugation if and only if G is either compact or is isomorphic to a direct product of a compact group and a vector group. Some of our results generalise those of Shah and Yadav.

In sharp contrast to the Hardy space case, the algebraic properties of Toeplitz operators on the Bergman space are quite different and abnormally complicated. In this paper, we study the finite-rank problem for a class of operators consisting of all finite linear combinations of Toeplitz products with monomial symbols on the Bergman space of the unit disk. It turns out that such a problem is equivalent to the problem of when the corresponding finite linear combination of rational functions is zero. As an application, we consider the finite-rank problem for the commutator and semi-commutator of Toeplitz operators whose symbols are finite linear combinations of monomials. In particular, we construct many motivating examples in the theory of algebraic properties of Toeplitz operators.

We show that the $C^*$-algebra of a row-finite source-free k-graph is Rieffel–Morita equivalent to a crossed product of an approximately finite-dimensional (AF) algebra by the fundamental group of the k-graph. When the k-graph embeds in its fundamental groupoid, this AF algebra is a Fell algebra; and simple-connectedness of a certain sub-1-graph characterises when this Fell algebra is Rieffel–Morita equivalent to a commutative $C^*$-algebra. We provide a substantial suite of results for determining if a given k-graph embeds in its fundamental groupoid, and provide a large class of examples, arising via work of Cartwright et al. [‘Groups acting simply transitively on the vertices of a building of type $\tilde{\rm A}_2$ I’, Geom. Dedicata 47 (1993), 143–166], Cartwright et al. ‘Groups acting simply transitively on the vertices of a building of type $\tilde{\rm A}_2$ II’, Geom. Dedicata 47 (1993), 167–226] and Robertson and Steger [‘Affine buildings, tiling systems and higher rank Cuntz–Krieger algebras’, J. reine angew. Math. 513 (1999), 115–144] from the theory of $\tilde {A_2}$-groups, which do embed.

We show that the fundamental groups of smooth $4$-manifolds that admit geometric decompositions in the sense of Thurston have asymptotic dimension at most four, and equal to four when aspherical. We also show that closed $3$-manifold groups have asymptotic dimension at most three. Our proof method yields that the asymptotic dimension of closed $3$-dimensional Alexandrov spaces is at most three. Thus, we obtain that the Novikov conjecture holds for closed $4$-manifolds with such a geometric decomposition and for closed $3$-dimensional Alexandrov spaces. Consequences of these results include a vanishing result for the Yamabe invariant of certain $0$-surgered geometric $4$-manifolds and the existence of zero in the spectrum of aspherical smooth $4$-manifolds with a geometric decomposition.

The well-known $abc$-conjecture concerns triples $(a,b,c)$ of nonzero integers that are coprime and satisfy ${a+b+c=0}$. The strong n-conjecture is a generalisation to n summands where integer solutions of the equation ${a_1 + \cdots + a_n = 0}$ are considered such that the $a_i$ are pairwise coprime and satisfy a certain subsum condition. Ramaekers studied a variant of this conjecture with a slightly different set of conditions. He conjectured that in this setting the limit superior of the so-called qualities of the admissible solutions equals $1$ for any n. In this paper, we follow results of Konyagin and Browkin. We restrict to a smaller, and thus more demanding, set of solutions, and improve the known lower bounds on the limit superior: for ${n \geq 6}$ we achieve a lower bound of $\frac 54$; for odd $n \geq 5$ we even achieve $\frac 53$. In particular, Ramaekers’ conjecture is false for every ${n \ge 5}$.

In this paper we study deformations of $C^*$-algebras that are given as cross-sectional $C^*$-algebras of Fell bundles $\mathcal A$ over locally compact groups G. Our deformation comes from a direct deformation of the Fell bundles $\mathcal A$ via certain parameters, such as automorphisms of the Fell bundle, group cocycles, or central group extensions of G by the circle group $\mathbb T$, and then taking cross-sectional algebras of the deformed Fell bundles. We then show that this direct deformation method is equivalent to the deformation via the dual coactions by similar parameters as studied previously in [4, 7].

In this article, we study the behavior of complete two-sided hypersurfaces immersed in the hyperbolic space $\mathbb H^{n+1}$. Initially, we introduce the concept of the linearized curvature function $\mathcal {F}_{r,s}$ of a two-sided hypersurface, its associated modified Newton transformation $\mathcal {P}_{r,s}$ and its naturally attached differential operator $\mathcal {L}_{r,s}$. Then, we obtain two formulas for differential operator $\mathcal {L}_{r,s}$ acting on the height function of a two-sided hypersurface and, for the case where their support functions are related by a negative constant, we derive two new formulas for the Newton transformation $P_{r}$ and the modified Newton transformation $\mathcal {P}_{r,s}$ acting on a gradient of the height function. Finally, these formulas, jointly with suitable maximum principles, enable us to establish our rigidity and nonexistence results concerning complete two-sided hypersurfaces in $\mathbb H^{n+1}$.

In this paper, we show that the diffraction of the primes is absolutely continuous, showing no bright spots (Bragg peaks). We introduce the notion of counting diffraction, extending the classical notion of (density) diffraction to sets of density zero. We develop the counting diffraction theory and give many examples of sets of zero density of all possible spectral types.

Let $\mu _{M,D}$ be the self-similar measure generated by $M=RN^q$ and the product-form digit set $D=\{0,1,\ldots ,N-1\}\oplus N^{p_1}\{0,1,\ldots ,N-1\}\oplus \cdots \oplus N^{p_s}\{0,1,\ldots ,N-1\}$, where $R\geq 2$, $N\geq 2$, q, $p_i(1\leq i\leq s)$ are integers with $\gcd (R,N)=1$ and $1\leq p_1<p_2<\cdots <p_s<q$. In this paper, we first show that $\mu _{M,D}$ is a spectral measure with a model spectrum $\Lambda $. Then, we completely settle two types of spectral eigenvalue problems for $\mu _{M,D}$. In the first case, for a real t, we give a necessary and sufficient condition under which $t\Lambda $ is also a spectrum of $\mu _{M,D}$. In the second case, we characterize all possible real numbers t such that $\Lambda '\subset \mathbb {R}$ and $t\Lambda '$ are both spectra of $\mu _{M,D}$.

In the early 1900s, Maillet [Introduction a la theorie des nombres transcendants et des proprietes arithmetiques des fonctions (Gauthier–Villars, Paris, 1906)] proved that the image of any Liouville number under a rational function with rational coefficients is again a Liouville number. The analogous result for quadratic Liouville matrices in higher dimensions turns out to fail. In fact, using a result by Kleinbock and Margulis [‘Flows on homogeneous spaces and Diophantine approximation on manifolds’, Ann. of Math. (2) 148(1) (1998), 339–360], we show that among analytic matrix functions in dimension $n\ge 2$, Maillet’s invariance property is only true for Möbius transformations with special coefficients. This implies that the analogue in higher dimensions of an open question of Mahler on the existence of transcendental entire functions with Maillet’s property has a negative answer. However, extending a topological argument of Erdős [‘Representations of real numbers as sums and products of Liouville numbers’, Michigan Math. J. 9 (1962), 59–60], we prove that for any injective continuous self-mapping on the space of rectangular matrices, many Liouville matrices are mapped to Liouville matrices. Dropping injectivity, we consider setups similar to Alniaçik and Saias [‘Une remarque sur les $G_{\delta }$-denses’, Arch. Math. (Basel) 62(5) (1994), 425–426], and show that the situation depends on the matrix dimensions $m,n$. Finally, we discuss extensions of a related result by Burger [‘Diophantine inequalities and irrationality measures for certain transcendental numbers’, Indian J. Pure Appl. Math. 32 (2001), 1591–1599] to quadratic matrices. We state several open problems along the way.

We show that the dynamical system associated by Putnam to a pair of graph embeddings is identical to the shift map on the limit space of a self-similar groupoid action on a graph. Moreover, performing a certain out-split on said graph gives rise to a Katsura–Exel–Pardo groupoid action on the out-split graph whose associated limit space dynamical system is conjugate to the previous one. We characterise the self-similar properties of these groupoids in terms of properties of their defining data, two matrices A, B. We prove a large class of the associated limit spaces are bundles of circles and points that fibre over a totally disconnected space, and the dynamics restricted to each circle are of the form $z\to z^{n}$. Moreover, we find a planar embedding of these spaces, thereby answering a question Putnam posed in his paper.

We prove that every coaction of a compact group on a finite-dimensional $C^*$-algebra is associated with a Fell bundle. Every coaction of a compact group on a matrix algebra is implemented by a unitary operator. A coaction of a compact group on $M_n$ is inner if and only if its fixed-point algebra has an abelian $C^*$-subalgebra of dimension n. Investigating the existence of effective ergodic coactions on $M_n$ reveals that $\operatorname {SO}(3)$ has them, while $\operatorname {SU}(2)$ does not. We give explicit examples of the two smallest finite nonabelian groups having effective ergodic coactions on $M_n$.

In the setting of product systems over group-embeddable monoids, we consider nuclearity of the associated Toeplitz C*-algebra in relation to nuclearity of the coefficient algebra. Our work goes beyond the known cases of single correspondences and compactly aligned product systems over right least common multiple (LCM) monoids. Specifically, given a product system over a submonoid of a group, we show, under technical assumptions, that the fixed-point algebra of the gauge action is nuclear if and only if the coefficient algebra is nuclear; when the group is amenable, we conclude that this happens if and only if the Toeplitz algebra itself is nuclear. Our main results imply that nuclearity of the Toeplitz algebra is equivalent to nuclearity of the coefficient algebra for every full product system of Hilbert bimodules over abelian monoids, over $ax+b$-monoids of integral domains and over Baumslag–Solitar monoids $BS^+(m,n)$ that admit an amenable embedding, which we provide for m and n relatively prime.