Joint Spectra of Generators in Topological Algebras

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.