A class of weighted convolution Fréchet algebras
Tom 91 / 2010
Streszczenie
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 $n\in\mathbb N$ there exists $m\in\mathbb N$ such that $\omega_m(t)/\omega_n(t)\to\infty$ as $t\to\infty$. Moreover, we characterise the continuous derivations on this algebra: Let $M(\omega_n)$ be the corresponding weighted measure algebras and let $B(\omega)=\bigcap_nM(\omega_n)$. If for all $n\in\mathbb N$ there exists $m\in\mathbb N$ such that $t\omega_n(t)/\omega_m(t)$ is bounded on $\mathbb R^+$, then the continuous derivations on $A(\omega)$ are exactly the linear maps $D$ of the form $D(f)=(Xf)*\mu$ for $f\in A(\omega)$, where $\mu\in B(\omega)$ and $(Xf)(t)=tf(t)$ for $t\in\mathbb R^+$ and $f\in A(\omega)$. If the condition is not satisfied, we show that $A(\omega)$ has no non-zero derivations.
