For an increasing sequence $(\omega_n)$ of algebra weights on $\mathbb R^+$
we study various properties of the Fr\'{e}chet 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 $t\to\infty$.