A class of weighted convolution Fréchet algebras

Abstract

For an increasing sequence $({\omega }_n)$ of algebra weights on ${\mathbb{R}}^{+}$ we study various properties of the Fréchet algebra $A(\omega )={\bigcap }_n{L}^1({\omega }_n)$ obtained as the intersection of the weighted Banach algebras $L^1({\omega }_n)$. We show that every endomorphism of $A(\omega )$ is standard, if for all n\in\mathbb N$thereexists$m\in\mathbb N$suchthat$ω_m(t)/ω_n(t)\to\infty$as$t\to\infty$.Moreover,wecharacterisethecontinuousderivationsonthisalgebra:Ifforall$n\in\mathbb N$thereexists$m\in\mathbb N$suchthat$t*ω_n(t)/ω_m(t)$isboundedon$\mathbb R^+$,thenthecontinuousderivationson$A(ω)$areexactlythelinearmaps$D$oftheform$D(f)=(Xf)*μ$for$f\in A(ω)$,where$μ$isameasurein$B(ω)=\bigcap_n M(ω_n)$and$(Xf)(t)=tf(t)$for$t\in\mathbb R^+$and$f\in A(ω)$.Iftheconditionisnotsatisfied,weshowthat$A(ω)$ has no non-zero derivations.

14 pages