LOCALLY PSEUDOCONVEX INDUCTIVE LIMIT 9 for every a A. When spA(a) = {ϕ(a) : ϕ hom A} for each a A, then (A, τ) is called a topological algebra with functional spectrum or (A, τ) has functional spectrum (see [2]). In this case ρA(a) = sup ϕ∈homA | ϕ(a) | . Therefore, in this case ρA is a seminorm on A. It is known (see [2, pp. 22–23]) that every commutative unital invertibly complete9 locally m-pseudoconvex Hausdorff algebra over C has functional spectrum. Moreover, there are given equivalent con- ditions for a commutative unital locally m-pseudoconvex algebra to be a topological algebra with functional spectrum. Lemma 2.3. Let (E, τ) be a topological algebra. Then, the spectral radius ρE is upper semicontinuous on (E, τ) if and only if (E, τ) is a Q-algebra. Proof. See [3, p. 66] and for unital algebras see [5, p. 118], [6, pp. 159–160], [7, p. 13] or [12, p. 60]. Proposition 2.4. 1. Let E be a topological algebra over C such that E = −→ Eα, where are topological algebras over C. a) If spE(x) and spEα (x) (at least for one) α I are proper subsets of C, then for such x E are true10 (1) spE(x) = α∈Ix spEα (x) (2) ρE(x) inf α∈Ix ρEα (x). Moreover, if (0,ρEα (x)) spEα (x) for every α Ix, then ρE(x) = inf α∈Ix ρEα (x) b) QinvE = α∈I QinvEα and InvE = α∈I InvEα, if E is a unital algebra. 2. Let (E, τ) be a locally pseudoconvex algebra over C such that E = −→ Eα, where every (Eα,τα) is a locally pseudoconvex Q-algebra and τ the locally pseudo- convex inductive limit topology on E. If, in addition, one of the following statements holds: (3) QinvEα τ for each α I (4) the spectral radius ρEα of is a seminorm on for each α I (5) I has the minimal element α0 and fβα0 is an open map for each β I, then (E, τ) is a Q-algebra. Proof. To see that the statement (1) holds, let first λ = 0 be a complex number and λ α∈Ix spEα (x). 9A topological algebra (A, τ) is invertibly complete (see [2, p. 15]) if every invertibly con- vergent Cauchy net (aλ)λ∈Λ (that is, there exists an element a A such that (aaλ)λ∈Λ and (aλa)λ∈Λ converge to eA) converges in A. It is known (see [15, p. 45 ]) that every complete unital algebra and every unital Q-algebra are invertibly complete. 10Here Ix = I : x Eα} for each x E.
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