International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 96289, 7 pages
doi:10.1155/2007/96289
Research Article
Joint Spectra of Generators in Topological Algebras
Departamento de Matemáticas, Universidad Autónoma Metropolitana, Unidad Iztapalapa, Avenue San Rafael Atlixco 186, Col. Vicentina, AP 55-534, México 09340, DF, Mexico
Received 29 October 2006; Revised 24 January 2007; Accepted 8 February 2007
Academic Editor: Ricardo Estrada
Copyright © 2007 Antoni Wawrzyńczyk. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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
σ𝒰(S)={(λs)s∈S∈ℂS | ∃ I ∈𝒰(s−λs)∈I ∀s∈S}. We prove that for S generating 𝒜 the spectrum σ𝒰(S) can be identified with the set 𝔐𝒰 of
continuous multiplicative functionals f on 𝒜 such that ker f∈𝒰. The relation is given by the formula
σ𝒰(S)={(f(s))s∈S | f∈𝔐𝒰}. If 𝒜 is a Q-algebra, the set σ𝒰(S) is rationally convex.