Let $𝒜$ be a commutative complex semisimple Banach algebra. Denote by $\mathrm{kh}\left(\mathrm{soc}\right(𝒜\left)\right)$ the kernel of the hull of the socle of $𝒜$. In this work we give some new characterizations of this ideal in terms of minimal idempotents in $𝒜$. This allows us to show that a “result” from Riesz theory in commutative Banach algebras is not true.

Let A be a commutative Banach algebra and let ${\Sigma }_A$ be its structure space. The norm spectrum σ(f) of the functional f ∈ A* is defined by $\sigma \left(f\right)=\overline{f·a:a\in A}\cap {\Sigma }_A$, where f·a is the functional on A defined by ⟨f·a,b⟩ = ⟨f,ab⟩, b ∈ A. We investigate basic properties of the norm spectrum in certain classes of commutative Banach algebras and present some applications.

An example is given of a semisimple commutative Banach algebra that has the strong spectral extension property but fails the multiplicative Hahn-Banach property. This answers a question posed by M. J. Meyer in [4].

Nous donnons dans ce travail une caractérisation des algèbres (semi-simples) localement-convexes complètes faiblement topologisées au sens de S. Warner, ce qui clarifie, entre autres, plusiers résultats données sur certaines classes d'algèbres à base étudiées par de nombreux auteurs ([2], [6], [7]) pour approcher le problème de E. A. Michael sur la continuité des caractères dans les algèbres de Fréchet [9].

The aim of this paper is an investigation of topological algebras with an orthogonal sequence which is total. Closed prime ideals or closed maximal ideals are kernels of multiplicative functionals and the continuous multiplicative functionals are given by the “coefficient functionals”. Our main result states that an orthogonal total sequence in a unital Fréchet algebra is already a Schauder basis. Further we consider algebras with a total sequence ${\left(x_n\right)}_{n\in \mathbb{N}}$ satisfying $x_n^2=x_n$ and $x_n{x}_{n+1}=x_{n+1}$ for all n ∈ ℕ.

It is shown that all maximal regular ideals in a Hausdorff topological algebra A are closed if the von Neumann bornology of A has a pseudo-basis which consists of idempotent and completant absolutely pseudoconvex sets. Moreover, all ideals in a unital commutative sequentially Mackey complete Hausdorff topological algebra A with jointly continuous multiplication and bounded elements are closed if the von Neumann bornology of A is idempotently pseudoconvex.

Soit $A$ une algèbre uniforme et soit $I$ un idéal fermé de $A$ tel que $A/I$ soit une algèbre isométriquement isomorphe à $C\left(X\right)$, il existe alors une sous-algèbre fermée $B\subset A$ telle que $A$ est isométriquement isomorphe à $I\oplus B$.

We consider the quantization of inversion in commutative p-normed quasi-Banach algebras with unit. The standard questions considered for such an algebra A with unit e and Gelfand transform x ↦ x̂ are: (i) Is $K_{\nu }=sup{\left|\right|(e-x)}^{-1}{\left|\right|}_p{:x\in A,\left|\right|x\left|\right|}_p\le 1,max\left|x̂\right|\le \nu$ bounded, where ν ∈ (0,1)? (ii) For which δ ∈ (0,1) is $C_{\delta }=sup\left|\right|x^{-1}{\left|\right|}_p{:x\in A,\left|\right|x\left|\right|}_p\le 1,min\left|x̂\right|\ge \delta$ bounded? Both questions are related to a “uniform spectral radius” of the algebra, $r_{\infty }\left(A\right)$, introduced by Björk. Question (i) has an affirmative answer if and only if $r_{\infty }\left(A\right)<1$, and this result is extended to more general nonlinear extremal problems...

A unital commutative Banach algebra is spectrally separable if for any two distinct non-zero multiplicative linear functionals φ and ψ on it there exist a and b in such that ab = 0 and φ(a)ψ(b) ≠ 0. Spectrally separable algebras are a special subclass of strongly harmonic algebras. We prove that a unital commutative Banach algebra is spectrally separable if there are enough elements in such that the corresponding multiplication operators on have the decomposition property (δ). On the other hand,...

Let ω be a weight on an LCA group G. Let M(G,ω) consist of the Radon measures μ on G such that ωμ is a regular complex Borel measure on G. It is proved that: (i) M(G,ω) is regular iff M(G,ω) has unique uniform norm property (UUNP) iff L¹(G,ω) has UUNP and G is discrete; (ii) M(G,ω) has a minimum uniform norm iff L¹(G,ω) has UUNP; (iii) M₀₀(G,ω) is regular iff M₀₀(G,ω) has UUNP iff L¹(G,ω) has UUNP, where M₀₀(G,ω) := {μ ∈ M(G,ω) : μ̂ = 0 on Δ(M(G,ω))∖Δ(L¹(G,ω))}.