Related papers: Strong 1-boundedness, $L^2$-Betti numbers, algebra…

In this note, we developed several results concerning abelian von Neumann algebras, their spectrums, and their tensor products with other von Neumann algebras. In particular, we developed a theory connecting elements of the spectrum of…

This article presents a method for proving upper bounds for the first $\ell^2$-Betti number of groups using only the geometry of the Cayley graph. As an application we prove that Burnside groups of large prime exponent have vanishing first…

We prove a Kunneth formula computing the Connes-Shlyakhtenko L^2-Betti numbers of the algebraic tensor product of two tracial *-algebras in terms of the L^2-Betti numbers of the two original algebras. As an application, we construct…

We study the space of traces associated with arbitrary full free products of unital, separable $C^*$-algebras. We show that, unless certain basic obstructions (which we fully characterize) occur, the space of traces always results in the…

Ho proved in [A note on the total domination number, Util.Math. 77 (2008) 97--100] that the total domination number of the Cartesian product of any two graphs with no isolated vertices is at least one half of the product of their total…

A new approach to Nori's weak Lefschetz theorem is described. The new approach, which involves the dbar-method, avoids moving arguments and gives much stronger results. In particular, it is proved that if X and Y are connected smooth…

Dujmovi\'{c}, Joret, Micek, Morin, Ueckerdt, and Wood [J. ACM 2020] established that every planar graph is a subgraph of the strong product of a graph with bounded treewidth and a path. Motivated by this result, this paper systematically…

We show that the amalgamated free product of weakly exact von Neumann algebras is weakly exact. This is done by using a universal property of Toeplitz-Pimsner algebras and a locally convex topology on bimodules of von Neumann algebras,…

Let $\Fth$ be a single vertex \textsf{k}-graph, and $\pi_\omega(\O_\theta)"$ be the von Neumann algebra induced from the GNS representation of a distinguished state $\omega$ of its $\textsf{k}$-graph C*-algebra $\O_\theta$. In this paper,…

We prove several analogs of Gromov's macroscopic dimension conjecture with extra curvature assumptions. More explicitly, we show that for an open Riemannian $n$-manifold $(M,g)$ of nonnegative Ricci (resp. sectional) curvature, if it has…

The category of $C^*$-algebras is blessed with many different tensor products. In contrast, virtually the only tensor product ever used in the category of von Neumann algebras is the normal spatial tensor product. We propose a definition of…

The $\{K_2,C_n\}$-factor of a graph is a spanning subgraph whose each component is either $K_2$ or $C_n$. In this paper, a sufficient condition with regard to tight toughness, isolated toughness and binding number bounds to guarantee the…

The strong geodetic problem is a recent variation of the geodetic problem. For a graph $G$, its strong geodetic number ${\rm sg}(G)$ is the cardinality of a smallest vertex subset $S$, such that each vertex of $G$ lies on a fixed shortest…

Let $A$ be a finite dimensional unital commutative associative algebra and let $B$ be a finite dimensional vertex $A$-algebroid such that its Levi factor is isomorphic to $sl_2$. Under suitable conditions, we construct an indecomposable…

Let $G$ be a connected graph. A vertex $w\in V(G)$ strongly resolves two vertices $u,v\in V(G)$ if there exists some shortest $u-w$ path containing $v$ or some shortest $v-w$ path containing $u$. A set $S$ of vertices is a strong metric…

In this paper we study non-nilpotent non-Lie Leibniz $\mathbb{F}$-algebras with one-dimensional derived subalgebra, where $\mathbb{F}$ is a field with $\operatorname{char}(\mathbb{F}) \neq 2$. We prove that such an algebra is isomorphic to…

In this paper, we exhibit strongly singular maximal abelian subalgebras living inside certain k-folded tensors of von Neumann group factors. The two classes of groups under consideration are the free groups of rank greater than 2 and the…

We compute the $L^2$-Betti numbers of the free $C^*$-tensor categories, which are the representation categories of the universal unitary quantum groups $A_u(F)$. We show that the $L^2$-Betti numbers of the dual of a compact quantum group…

Anari, Gharan, and Vinzant proved (complete) log-concavity of the basis generating functions for all matroids. From the viewpoint of combinatorial Hodge theory, it is natural to ask whether these functions are "strictly" log-concave for…

This article is dedicated to the study of the acylindrical hyperbolicity of automorphism groups of graph products of groups. Our main result is that, if $\Gamma$ is a finite graph which contains at least two vertices and is not a join and…