For an increasing sequence (ωₙ) of algebra weights on ℝ⁺ we study various properties of the Fréchet algebra A(ω) = ⋂ ₙ L¹(ωₙ) obtained as the intersection of the weighted Banach algebras L¹(ωₙ). We show that every endomorphism of A(ω) is standard, if for all n ∈ ℕ there exists m ∈ ℕ such that as t → ∞. Moreover, we characterise the continuous derivations on this algebra: Let M(ωₙ) be the corresponding weighted measure algebras and let B(ω) = ⋂ ₙM(ωₙ). If for all n ∈ ℕ there exists m ∈ ℕ such that is bounded on ℝ⁺, then the continuous derivations on A(ω) are exactly the linear maps D of the form D(f) = (Xf)*μ for f ∈ A(ω), where μ ∈ B(ω) and (Xf)(t) = tf(t) for t ∈ ℝ⁺ and f ∈ A(ω). If the condition is not satisfied, we show that A(ω) has no non-zero derivations.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc91-0-15,
author = {Thomas Vils Pedersen},
title = {A class of weighted convolution Fr\'echet algebras},
journal = {Banach Center Publications},
volume = {89},
year = {2010},
pages = {247-259},
zbl = {1216.46048},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc91-0-15}
}
Thomas Vils Pedersen. A class of weighted convolution Fréchet algebras. Banach Center Publications, Tome 89 (2010) pp. 247-259. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc91-0-15/