Joint Spectra of Generators in Topological Algebras

Antoni Wawrzyńczyk

International Journal of Mathematics and Mathematical Sciences, 2007, vol. 2007, 1-7

Abstract:

Let 𝒜 be a complex topological algebra with unit 1 and 𝒰 a family of proper closed ideals in 𝒜 . For an arbitrary S ⊂ 𝒜 we define a globally defined joint spectrum σ 𝒰 ( S ) = { ( λ s ) s ∈ S ∈ ℂ S | ∃ I ∈ 𝒰 ( s − λ s ) ∈ I ∀ s ∈ S } . We prove that for S generating 𝒜 the spectrum σ 𝒰 ( S ) can be identified with the set 𝔐 𝒰 of continuous multiplicative functionals f on 𝒜 such that ker f ∈ 𝒰 . The relation is given by the formula σ 𝒰 ( S ) = { ( f ( s ) ) s ∈ S | f ∈ 𝔐 𝒰 } . If 𝒜 is a Q -algebra, the set σ 𝒰 ( S ) is rationally convex.

Date: 2007
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:096289

DOI: 10.1155/2007/96289

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