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Abstract: For an increasing sequence $\omega n$ of algebra weights on $\mathbb R^+$we study various properties of the Fr\-{e}chet algebra $A\omega=\bigcap nL^1\omega n$ obtained as the intersection of the weighted Banach algebras$L^1\omega n$. We show that every endomorphism of $A\omega$ is standard, iffor all n\in\mathbb N$ there exists $m\in\mathbb N$ such that$\omega mt-\omega nt\to\infty$ as $t\to\infty$. Moreover, we characterisethe continuous derivations on this algebra: If for all $n\in\mathbb N$ thereexists $m\in\mathbb N$ such that $t*\omega nt-\omega mt$ is bounded on$\mathbb R^+$, then the continuous derivations on $A\omega$ are exactly thelinear maps $D$ of the form $Df=Xf*\mu$ for $f\in A\omega$, where $\mu$is a measure in $B\omega=\bigcap n M\omega n$ and $Xft=tft$ for$t\in\mathbb R^+$ and $f\in A\omega$. If the condition is not satisfied, weshow that $A\omega$ has no non-zero derivations.



Author: Thomas Vils Pedersen

Source: https://arxiv.org/







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