A class of weighted convolution Fréchet algebras

Abstract

For an increasing sequence $(ω_n)$ of algebra weights on $\mathbb R^+$ we study various properties of the Fréchet algebra $A(ω)=\bigcap_n L^1(ω_n)$ obtained as the intersection of the weighted Banach algebras $L^1(ω_n)$. We show that every endomorphism of $A(ω)$ is standard, if for all n\in\mathbb N$ there exists $m\in\mathbb N$ such that $ω_m(t)/ω_n(t)\to\infty$ as $t\to\infty$. Moreover, we characterise the continuous derivations on this algebra: If for all $n\in\mathbb N$ there exists $m\in\mathbb N$ such that $t*ω_n(t)/ω_m(t)$ is bounded on $\mathbb R^+$, then the continuous derivations on $A(ω)$ are exactly the linear maps $D$ of the form $D(f)=(Xf)*μ$ for $f\in A(ω)$, where $μ$ is a measure in $B(ω)=\bigcap_n M(ω_n)$ and $(Xf)(t)=tf(t)$ for $t\in\mathbb R^+$ and $f\in A(ω)$. If the condition is not satisfied, we show that $A(ω)$ has no non-zero derivations.