Joint Spectra of Generators in Topological Algebras

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