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Compactness and weak-star continuity of derivations on weighted convolution algebras. (English) Zbl 1256.43001

Let \(\omega:\mathbb{R}^+\to (0,\infty)\) be a continuous weight (i.e., \(\omega(0)=1\) and \(\omega(s+t)\leq \omega(s)\omega(t)\) for all \(s,t\in\mathbb{R}^+\)). Results of Grønbæk and Bade & Dales show that the continuous derivations from the weighted convolution algebra \(L^1(\omega)\) into its dual \(L^\infty(1/\omega)\) are precisely the maps of the form \[ (D_\varphi f)(t)=\int_0^\infty f(s)\frac{s}{t+s}\varphi(t+s)ds\quad(t\in\mathbb{R}^+\,,\;f\in L^1(\omega)\,. \] Moreover, every such derivation extends uniquely to a continuous derivation \(\bar{D}_\varphi:M(\omega)\to L^\infty(1/\omega)\), where \(M(\omega)\) is the weighted measure algebra. In this paper, the author proves sufficient conditions on \(\varphi\) under which (i) \(\bar{D}_\varphi\) is weak-star continuous; (ii) \(\bar{D}_\varphi\) is compact; and (iii) \(D_\varphi\) is compact.

MSC:

43A20 \(L^1\)-algebras on groups, semigroups, etc.
46J10 Banach algebras of continuous functions, function algebras
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References:

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