相关论文: On the Dales-Zelazko conjecture for Beurling algeb…

Let $G$ be a locally compact group. We prove that the augmentation ideal in $L^1(G)$ is (algebraically) finitely-generated as a left ideal if and only if $G$ is finite. We then investigate weighted versions of this result, as well as a…

We characterize $C^*$-algebras and $C^*$-modules such that every maximal right ideal (resp. right submodule) is algebraically finitely generated. In particular, $C^*$-algebras satisfy the Dales--\.Zelazko conjecture.

The purpose of this note is to describe some algebraic conditions on a Banach algebra which force it to be finite dimensional. One of the main results in Theorem~2 which states that for a locally compact group $G$, $G$ is compact if there…

Let $G$ be a locally compact abelian group, and let $\omega:G \to [1,\infty)$ be a measurable weight, i.e., $\omega$ is measurable, and $\omega(s+t)\leq \omega(s)\omega(t)$ for all $s, t \in G$. Let $\mathcal{A}$ be a semisimple commutative…

In 1971, Grauert and Remmert proved that a commutative, complex, Noetherian Banach algebra is necessarily finite-dimensional. More precisely, they proved that a commutative, complex Banach algebra has finite dimension over $\C$ whenever all…

It is an open problem whether an infinite-dimensional amenable Banach algebra exists whose underlying Banach space is reflexive. We give sufficient conditions for a reflexive, amenable Banach algebra to be finite-dimensional (and thus a…

We prove that for a Banach algebra $A$ having a bounded $\mathcal{Z}(A)$-approximate identity and for every $\bf[IN]$ group $G$ with weight $w$ which is either constant on conjugacy classes or $w \geq 1$, $\mathcal{Z}\big(L^1_w(G)…

We study the class of Bernstein algebras that are algebraic, in the sense that each element generates a finite-dimensional subalgebra. Every Bernstein algebra has a maximal algebraic ideal, and the quotient algebra is a zero-multiplication…

Let $G$ be a locally compact abelian group, and let $\omega:G \to [1,\infty)$ be a weight, i.e., $\omega$ is measurable, $\omega$ is locally bounded and $\omega(s+t)\leq \omega(s)\omega(t)$ for all $s, t \in G$. If $\omega^{-1}$ is…

We prove that the bidual of a Beurling algebra on $\mathbb{Z}$, considered as a Banach algebra with the first Arens product, can never be semisimple. We then show that ${\rm rad\,}(\ell^{\, 1}(\oplus_{i=1}^\infty \mathbb{Z})")$ contains…

In this paper, we study weak amenability of Beurling algebras. To this end, we introduce the notion inner quasi-additive functions and prove that for a locally compact group $G$, the Banach algebra $L^1(G, \omega)$ is weakly amenable if and…

Let $L^1_\om(G)$ be a Beurling algebra on a locally compact abelian group $G$. We look for general conditions on the weight which allows the vanishing of continuous derivations of $L^1_\om(G)$. This leads us to introducing vector-valued…

We investigate ideals of the form $\{A \subseteq \omega\colon \sum_{n\in A} x_n$ is unconditionally convergent $\}$, where $(x_n)_{n\in\omega}$ is a sequence in a Polish group or in a Banach space. If an ideal on $\omega$ can be seen in…

We formulate and prove a version of the Segal Conjecture for infinite groups. For finite groups it reduces to the original version. The condition that G is finite is replaced in our setting by the assumption that there exists a finite model…

We address the following two questions regarding the maximal left ideals of the Banach algebra $\mathscr{B}(E)$ of bounded operators acting on an infinite-dimensional Banach pace $E$: (Q1) Does $\mathscr{B}(E)$ always contain a maximal left…

We consider a family of finitely presented groups, called Universal Left Invertible Element (or ULIE) groups, that are universal for existence of one--sided invertible elements in a group ring K[G], where K is a field or a division ring. We…

The finitistic dimension conjecture asserts that any finite-dimensional algebra over a field should have finite finitistic dimension. Recently, this conjecture is reduced to studying finitistic dimensions for extensions of algebras. In this…

We classify the weak*-closed maximal left ideals of the measure algebra $M(G)$ for certain Hermitian locally compact groups $G$ in terms of the irreducible representations of $G$ and their asymptotic properties. In particular, we obtain a…

One of the classic results of group theory is the so-called Schur theorem. It states that if the central factor-group $G/\zeta(G)$ of a group $G$ is finite, then its derived subgroup $[G,G]$ is also finite. This result has numerous…

We show that the nuclear dimension of a (twisted) group C*-algebra of a virtually polycyclic group is finite. This prompts us to make a conjecture relating finite nuclear dimension of group C*-algebras and finite Hirsch length, which we…