Related papers: The problem of zero divisors in convolution algebr…

We establish the absence of zero divisors in the reduction algebra of a Lie algebra g with respect to its reductive Lie sub-algebra k. The class of reduction algebras include the Lie algebras (they arise when k is trivial) and the…

A famous conjecture about group algebras of torsion-free groups states that there is no zero divisor in such group algebras. A recent approach to settle the conjecture is to show the non-existence of zero divisors with respect to the length…

We associate a graph to a possible non-zero zero-divisor in the group algebra of a torsion-free group.

Consider the Kontsevich $\star$-product on the symmetric algebra of a finite dimensional Lie algebra $\mathfrak g$, regarded as the algebra of distributions with support 0 on $\mathfrak g$. In this paper, we extend this $\star$-product to…

Kaplansky's zero divisor conjecture (unit conjecture, respectively) states that for a torsion-free group $G$ and a field $\mathbb{F}$, the group ring $\mathbb{F}[G]$ has no zero divisors (has no unit with support of size greater than $1$).…

Let S be a standard graded Artinian algebra over a field k. We identify constraints on the Hilbert function of S which are imposed by the hypothesis that S contains an exact pair of homogeneous zero divisors. As a consequence, we prove that…

This paper addresses two of Kaplansky's conjectures concerning group rings $K[G]$, where $K$ is a field and $G$ is a torsion-free group: the zero-divisor conjecture, which asserts that $K[G]$ has no non-trivial zero-divisors, and the unit…

We prove that any countable discrete and torsion free subgroup of a general linear group over an arbitrary field or a similar subgroup of an almost connected Lie group satisfies the integral algebraic K-theoretic (split) Novikov conjecture…

Kaplanski's Zero Divisor Conjecture envisions that for a torsion-free group G and an integral domain R, the group ring R[G] does not contain non-trivial zero divisors. We define the length of an element a in R[G] as the minimal non-negative…

We consider the 2-generated free metabelian associative and Lie algebras over the complex field and the invariants of the dihedral groups of finite order acting on these algebras. In the associative case we find a finite set of generators…

The concept of a C-approximable group, for a class of finite groups C, is a common generalization of the concepts of a sofic, weakly sofic, and linear sofic group. Glebsky raised the question whether all groups are approximable by finite…

In this note we prove that in the case of finitely generated amenable groups the classical zero divisor conjecture implies the analytic zero divisor conjecture of Linnell.

The Titchmarsh convolution theorem is a celebrated result about the support of the convolution of two functions. We present a simple proof based on the canonical factorization theorem for bounded holomorphic functions on the unit disk.

A famous theorem of Dixmier-Malliavin asserts that every smooth, compactly-supported function on a Lie group can be expressed as a finite sum in which each term is the convolution, with respect to Haar measure, of two such functions. We…

Let $k$ be a division ring and let $G$ be either a torsion-free virtually compact special group or a finitely generated torsion-free $3$-manifold group. We embed the group algebra $kG$ in a division ring and prove that the embedding is…

In this paper we concern with positive zero divisors in $C^{*}$ algebras. By means of zero divisors, we introduce a hereditary invariant for $C^{*}$ algebras. Using this invariant, we give an example of a $C^{*}$ algebra $A$ and a $C^{*}$…

We show that the convolution algebra of smooth, compactly-supported functions on a Lie groupoid is H-unital in the sense of Wodzicki. We also prove H-unitality of infinite order vanishing ideals associated to invariant, closed subsets of…

We present a Suffridge-like extension of the Grace-Szeg\"o convolution theorem for polynomials and entire functions with only real zeros. Our results can also be seen as a $q$-extension of P\'olya's and Schur's characterization of…

We study a notion of indecomposability in differential algebraic groups which is inspired by both model theory and differential algebra. After establishing some basic definitions and results, we prove an indecomposability theorem for…

In the first part of my talk I will explain a solution to the extension of Lie's problem on classification of "local continuous transformation groups of a finite-dimensional manifold" to the case of supermanifolds. (More precisely, the…