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Old and new results on Allan's GB*-algebras

Maria Fragoulopoulou, Atsushi Inoue, Klaus-Detlef Kürsten (2010)

Banach Center Publications

This is an expository paper on the importance and applications of GB*-algebras in the theory of unbounded operators, which is closely related to quantum field theory and quantum mechanics. After recalling the definition and the main examples of GB*-algebras we exhibit their most important properties. Then, through concrete examples we are led to a question concerning the structure of the completion of a given C*-algebra 𝓐₀[||·||₀], under a locally convex *-algebra topology τ, making the multiplication...

On a certain class of subspectra

Andrzej Sołtysiak (1991)

Commentationes Mathematicae Universitatis Carolinae

The aim of this paper is to characterize a class of subspectra for which the geometric spectral radius is the same and depends only upon a commuting n -tuple of elements of a complex Banach algebra. We prove also that all these subspectra have the same capacity.

On a class of inner maps

Edoardo Vesentini (2005)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let f be a continuous map of the closure Δ ¯ of the open unit disc Δ of C into a unital associative Banach algebra A , whose restriction to Δ is holomorphic, and which satisfies the condition whereby 0 ∉ σ ⁢ f ⁢ z ⊂ Δ ¯ for all z ∈ Δ and σ ⁢ f ⁢ z ⊂ ∂ ⁡ Δ whenever z ∈ ∂ ⁡ Δ (where σ ⁢ x is the spectrum of any x ∈ A ). One of the basic results of the present paper is that f is , that is to say, σ ⁢ f ⁢ z is then a compact subset of ∂ ⁡ Δ that does not depend on z for all z ∈ Δ ¯ . This fact will be applied to holomorphic self-maps of the open unit ball of some J * -algebra...

On a problem of Bertram Yood

Mart Abel, Mati Abel (2014)

Topological Algebra and its Applications

In 1964, Bertram Yood posed the following problem: whether the intersection of all closed maximal regular left ideals of a topological ring coincides with the intersection of all closed maximal regular right ideals of this ring. It is proved that these two intersections coincide for advertive and simplicial topological rings and, using this result, it is shown that the topological left radical and the topological right radical for every advertive and simplicial topological algebra coincide.

On a theorem of Vesentini

Gerd Herzog, Christoph Schmoeger (2004)

Studia Mathematica

Let 𝒜 be a Banach algebra over ℂ with unit 1 and 𝑓: ℂ → ℂ an entire function. Let 𝐟: 𝒜 → 𝒜 be defined by 𝐟(a) = 𝑓(a) (a ∈ 𝒜), where 𝑓(a) is given by the usual analytic calculus. The connections between the periods of 𝑓 and the periods of 𝐟 are settled by a theorem of E. Vesentini. We give a new proof of this theorem and investigate further properties of periods of 𝐟, for example in C*-algebras.

On Arens-Michael algebras which do not have non-zero injective ⨶-modules

A. Pirkovskii (1999)

Studia Mathematica

A certain class of Arens-Michael algebras having no non-zero injective topological ⨶-modules is introduced. This class is rather wide and contains, in particular, algebras of holomorphic functions on polydomains in ℂ n , algebras of smooth functions on domains in ℝ n , algebras of formal power series, and, more generally, any nuclear Fréchet-Arens-Michael algebra which has a free bimodule Koszul resolution.

On Cental Morphisms

Athanassios Kyriazis (1992)

Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

On certain products of Banach algebras with applications to harmonic analysis

Mehdi Sangani Monfared (2007)

Studia Mathematica

Given Banach algebras A and B with spectrum σ(B) ≠ ∅, and given θ ∈ σ(B), we define a product A × θ B , which is a strongly splitting Banach algebra extension of B by A. We obtain characterizations of bounded approximate identities, spectrum, topological center, minimal idempotents, and study the ideal structure of these products. By assuming B to be a Banach algebra in ₀(X) whose spectrum can be identified with X, we apply our results to harmonic analysis, and study the question of spectral synthesis,...

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