For all convolution algebras L 1[0, 1); L loc1 and A(ω) = ∩n L 1(ωn), the derivations are of the form D μ f = Xf * μ for suitable measures μ, where (Xf)(t) = tf(t). We describe the (weakly) compact as well as the (weakly) Montel derivations on these algebras in terms of properties of the measure μ. Moreover, for all these algebras we show that the extension of D μ to a natural dual space is weak-star continuous.

The problem of representability of quadratic functionals by sesquilinear forms is studied in this article in the setting of a module over an algebra that belongs to a certain class of complex Banach *-algebras with an approximate identity. That class includes C*-algebras as well as H*-algebras and their trace classes. Each quadratic functional acting on such a module can be represented by a unique sesquilinear form. That form generally takes values in a larger algebra than the given quadratic functional...

Let A be a Banach algebra, and let D : A → A be a (possibly unbounded) derivation. We are interested in two problems concerning the range of D: 1. When does D map into the (Jacobson) radical of A? 2. If [a,Da] = 0 for some a ∈ A, is Da necessarily quasinilpotent? We prove that derivations satisfying certain polynomial identities map into the radical. As an application, we show that if [a,[a,[a,Da]]] lies in the prime radical of A for all a ∈ A, then D maps into the radical. This generalizes a result...

Let $X$ be a Banach space, $ℬ\left(X\right)$ the algebra of bounded linear operators on $X$ and $(J,\parallel ·{\parallel }_J)$ an admissible Banach ideal of $ℬ\left(X\right)$. For $T\in ℬ\left(X\right)$, let $L_{J,T}$ and $R_{J,T}\in ℬ\left(J\right)$ denote the left and right multiplication defined by $L_{J,T}\left(A\right)=TA$ and $R_{J,T}\left(A\right)=AT$, respectively. In this paper, we study the transmission of some concepts related to recurrent operators between $T\in ℬ\left(X\right)$, and their elementary operators $L_{J,T}$ and $R_{J,T}$. In particular, we give necessary and sufficient conditions for $L_{J,T}$ and $R_{J,T}$ to be sequentially recurrent. Furthermore, we prove that $L_{J,T}$ is recurrent if and only...

In this paper we consider the regularity problem for the commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ where $b$ is a locally integrable function and $(R_j{)}_{1\le j\le n}$ are the Riesz transforms in the $n$-dimensional euclidean space ${ℝ}^n$. More precisely, we prove that these commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ are bounded from $L^p$ into the Besov space ${\dot{B}}_p^{s,p}$ for $1\<p\<+\infty$ and $0\<s\<1$ if and only if $b$ is in the $BMO$-Triebel-Lizorkin space ${\dot{F}}_{\infty }^{s,p}$. The reduction of our result to the case $p=2$ gives in particular that the commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ are bounded form $L^2$ into the Sobolev space ${\dot{H}}^s$ if and only if $b$ is in the $BMO$-Sobolev...

A scattered element of a Banach algebra is an element with at most countable spectrum. The set of all scattered elements is denoted by (). The scattered radical ${}_{sc}\left(\right)$ is the largest ideal consisting of scattered elements. We characterize in several ways central elements of modulo the scattered radical. As a consequence, it is shown that the following conditions are equivalent: (i) () + () ⊂ (); (ii) ()() ⊂ (); (iii) $[\left(\right),]{\subset }_{sc}\left(\right)$.

This is the second instalment of my previous paper with the same title, [1]. This paper consists of two different parts. The first part is devoted to improvements of the results developed in [1]. These improvements are described in section 0.1 below and developed in sections 1 to 5, and 9 to 10; they are in fact technically distinct from [1] and rely on a systematic use of microlocalisation in the context of Hörmander-Weyl calculus. These paragraphs can therefore be read quite independently from...