8 MATI ABEL AND REYNA MAR´ IA PEREZ-TISCARE˜´ NO In addition, let B be a commutative A-bimodule with separate continuous module multiplication. Then B × A = −→ (Bα × A) (where × A has the product topology τα for each α I) is Q-LFpg algebra in the locally pseudoconvex inductive limit topology τind. Indeed, (Bα × A, τα) is a locally pseudoconvex F -algebra for every α I (see [13, p. 59, Prop. 6] and [16, p. 6]). Moreover, B × A in the product topology τ and × A in the product topology τα are unital Q-algebras by Proposition 2.1 and the inclusion φα : × A (B × A, τ) is continuous for each α I since (U × V ) (Bα × A) = (U Bα) × V τα for every U τB and V τA (the topology of coincides with the topology, induced on by B). Hence τ τind and therefore B × A is a Q-algebra also in the topology τind. Consequently, B × A is a Q-LFpg-algebra. 2.2. Cases, when the inductive limit of topological algebras is a Q-algebra. 1. First we consider the case when the inductive limit of topological algebras is a Q-algebra in the inductive limit topology. Theorem 2.2. Let (E, τ) be a topological algebra, which is an inductive limit of Q-algebras (Eα,τα) with α I, and τ is the inductive limit topology on E. Then, (E, τ) is Q-algebra. Proof. It is known that (E, τ) is a Q-algebra if and only if QinvE is a neigh- borhood of zero in E (see, for example, [15, p. 44]). To prove that QinvE is a neighborhood of zero in (E, τ), let : E be the canonical map for α I. Since every (Eα,τα) is a Q-algebra, then for every α I there exists an open neighbourhood of zero in such that QinvEα. So fα(Oα) fα(QinvEα) QinvE for every α I. Hence E0 = α∈I fα(Oα) QinvE. Since −1(E0) for every α I, then −1(E0) is a neighborhood of zero in for every α I. Hence E0 is a neighborhood of zero in E. So QinvE is also a neighborhood of zero in (E, τ). 2. Let now A be an algebra over C and let a A. The spectrum spA(a) of a is defined by spA(a) = λ C \ {0} : a λ QinvA {0} and the spectral radius ρA(a) of a by ρA(a) = sup{|λ| : λ spA(a)}. For every topological algebra (A, τ) let hom A denote the set of all nontrivial con- tinuous linear multiplicative functionals on A. Then {ϕ(a) : ϕ homA} spA(a)
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