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A class of weighted convolution Fréchet algebras

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Pedersen T.

Banach Center Publications

2010

tom 91

nr 1

247-259

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 $ω_m(t)/ωₙ(t) → ∞$ 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 $tωₙ(t)/ω_m(t)$ 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.

Ograniczanie wyników

1 Banach Center Publications

1 Pedersen T.

1 2010

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A class of weighted convolution Fréchet algebras