Zero divisors and L^p(G), II

Existence of zeros for operators taking their values in the dual of a Banach space

We prove, under certain conditions, the existence of zeros for a weakly continuous operator on a paracompact topological space into the dual of a Banach space.

On the "Zero-Two" Law for Positive Contractions on Banach-Kantorovich Lattice $L^p(\nabla,μ)$

In the present paper we prove the "zero-two" law for positive contractions of lattices $L^p(\nabla,μ)$ of Banach-Kantorovich, constructed by the measure $μ$ with values in the ring of all measurable functions.

Fourier convolution operators with symbols equivalent to zero at infinity on Banach function spaces

We study Fourier convolution operators $W^0(a)$ with symbols equivalent to zero at infinity on a separable Banach function space $X(\mathbb{R})$ such that the Hardy-Littlewood maximal operator is bounded on $X(\mathbb{R})$ and on its associate space $X'(\mathbb{R})$. We show that the limit operators of $W^0(a)$ are all equal to zero.

Zero Lie product determined Banach algebras

A Banach algebra $A$ is said to be zero Lie product determined if every continuous bilinear functional $φ\colon A\times A\to \mathbb{C}$ with the property that $φ(a,b)=0$ whenever $a$ and $b$ commute is of the form $φ(a,b)=τ(ab-ba)$ for some $τ\in A^*$. In the first part of the paper we give some general remarks on this class of algebras. In the second part we consider amenable Banach algebras and show that all group algebras $L^1(G)$ with $G$ an amenable locally compact group are zero Lie product determined.

Zero Lie product determined Banach algebras, II

A Banach algebra $A$ is said to be zero Lie product determined if every continuous bilinear functional $φ\colon A\times A\to \mathbb{C}$ satisfying $φ(a,b)=0$ whenever $ab=ba$ is of the form $φ(a,b)=ω(ab-ba)$ for some $ω\in A^*$. We prove that $A$ has this property provided that any of the following three conditions holds: (i) $A$ is a weakly amenable Banach algebra with property $\mathbb{B}$ and having a bounded approximate identity, (ii) every continuous cyclic Jordan derivation from $A$ into $A^*$ is an inner derivation, (iii) $A$ is the algebra of all $n\times n$ matrices, where $n\ge 2$, …

On the analytic zero divisor conjecture of Linnell

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.

Strongly zero product determined Banach algebras

$C^*$-algebras, group algebras, and the algebra $\mathcal{A}(X)$ of approximable operators on a Banach space $X$ having the bounded approximation property are known to be zero product determined. We are interested in giving a quantitative estimate of this property by finding, for each Banach algebra $A$ of the above classes, a constant $α$ with the property that for every continuous bilinear functional $φ\colon A \times A\to\mathbb{C}$ there exists a continuous linear functional $ξ$ on $A$ such that \[ \sup_{\Vert a\Vert=\Vert b\Vert=1}\vertφ(a,b)-ξ(ab)\vert\le α\sup_{\mathclap{\substack{\Vert…

零 Jordan 积决定的 Banach 代数

Zero Jordan product determined Banach algebras

本文定义了零 Jordan 积决定的 Banach 代数，并证明了所有 C*-代数和 amenable 局部紧群的群代数均满足该性质，且讨论了相关应用。

Amenable groups of finite cohomological dimension and the zero divisor conjecture

We prove that every amenable group of cohomological dimension two whose integral group ring is a domain is solvable and investigate certain homological finiteness properties of groups that satisfy the analytic zero divisor conjecture and act on an acyclic CW-complex with amenable stabilisers.

Functional identities and zero Lie product determined Banach algebras

Three problems connecting functional identities to the recently introduced notion of a zero Lie product determined Banach algebra are discussed. The first one concerns commuting linear maps, the second one concerns derivations that preserve commutativity, and the third one concerns bijective commutativity preserving linear maps.

C^*代数的零因子图与零因子实秩

Zero-divisor graphs and zero-divisor real rank of $C^*$-algebras

该研究定义了C^*代数的正零因子图Γ_+(A)并分析其连通性与直径，还提出零因子实秩ZDRR(A)，证明其与实秩RR(A)的关系及在不同C^*代数中的取值，构造出ZDRR取任意非负整数的例子。

The problem of zero divisors in convolution algebras of supersolvable Lie groups

We prove a variant of the Titchmarsh convolution theorem for simply connected supersolvable Lie groups, namely we show that the convolution algebras of compactly supported continuous functions and compactly supported finite measures on such groups do not contain zero divisors. This can be also viewed as a topological version of the zero divisor conjecture of Kaplansky.

A Gaussian Radon Transform for Banach Spaces

We develop a Radon transform on Banach spaces using Gaussian measure and prove that if a bounded continuous function on a separable Banach space has zero Gaussian integral over all hyperplanes outside a closed bounded convex set in the Hilbert space corresponding to the Gaussian measure then the function is zero outside this set.

Zero product determined Banach algebras

Let $\mathcal{L}$ be a completely distributive commutative subspace lattice or a subspace lattice with two atoms, we use a unified approach to study the derivations, homomorphisms on $\mathrm{Alg} \mathcal{L}$. We verify that the multiplier algebra of $\mathrm{Alg} \mathcal{L}\cap \mathcal{K}(\mathcal{H})$ is isomorphic to $\mathrm{Alg} \mathcal{L}$ and $\mathrm{Alg} \mathcal{L}$ is zero product determined. For $T$ in $M_{n}(\mathbb{C})$, $n\geq 2$, we show that $\mathcal{A}_{T}$ is zero product determined if and only if every local derivation from $\mathcal{A}_{T}$ into any Banach $\mathcal{A…