### A ${B}_{0}$-algebra without generalized topological divisors of zero

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Let A be a commutative unital Fréchet algebra, i.e. a completely metrizable topological algebra. Our main result states that all ideals in A are closed if and only if A is a noetherian algebra

In the present note, we characterize the essential set $E$ of a function algebra $A$ defined on a compact Hausdorff space $X$ in terms of local properties of functions in $A$ at the points off $E$.

A question of Warner and Whitley concerning a nonunital version of the Gleason-Kahane-Żelazko theorem is considered in the context of nonnormed topological algebras. Among other things it is shown that a closed hyperplane M of a commutative symmetric F*-algebra E with Lindelöf Gel'fand space is a maximal regular ideal iff each element of M belongs to some closed maximal regular ideal of E.

Let G be a locally compact abelian group and M(G) its measure algebra. Two measures μ and λ are said to be equivalent if there exists an invertible measure ϖ such that ϖ*μ = λ. The main result of this note is the following: A measure μ is invertible iff |μ̂| ≥ ε on Ĝ for some ε > 0 and μ is equivalent to a measure λ of the form λ = a + θ, where a ∈ L¹(G) and θ ∈ M(G) is an idempotent measure.

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 ${\omega}_{m}\left(t\right)/\omega \u2099\left(t\right)\to \infty $ 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...

A constructive proof of the Beurling-Rudin theorem on the characterization of the closed ideals in the disk algebra A(𝔻) is given.