LOCALLY PSEUDOCONVEX INDUCTIVE LIMIT 7 for every x R. Hence ψ(x) + a eA + Wεn Inv(A). It is easy to check that the inverse of (ψ, a) is (φa,a−1), where φa : R A is a function, defined by φa(x) = −(ψ(x) + a)−1ψ(x)a−1 for every x R and a eA +W ε 2 n . Since the invertion and the multiplications in A by an element from the right and from the left are continuous, then φa is continuous for every a eA + W ε 2 n . Moreover, φa(x) = θA if and only ψ(x) = θA. Hence φa K(R,A). Therefore, (ψ, a)−1 exists in K(R,A)×A for any (ψ, a) V ε 2 n . So we have proved that (θ, eA) V ε 2 n Inv(K(R,A)×A). Consequently, (K(R,A)×A, τ ) is a Q-algebra. Thus, (K(R,A) × A, τ ) is also a Q-algebra. Moreover, U ε 2 n K(R,A [−i, i]) = ψ K(R,A [−i, i]) : qin(ψ) ε 2 is a neighborhood of zero in K(R,A [−i, i]) in the topology τi for every i. Therefore V ε 2 n (K(R,A [−i, i]) × A) is a neighborhood of (θ, eA) in K(R,A [−i, i]) × A. Similarly as above, (θ, eA) V ε 2 n (K(R,A [−i, i]) × A) Inv(K(R,A [−i, i]) × A). Hence K(R,A [−i, i]))×A is also a Q-algebra for every i I. Thus, (K(R,A)×A, τ ) is a Q-LFp-algebra. To give other examples, we use the following result. Proposition 2.1. Let A be an unital Q-algebra and B a Ql-algebra6 which is an A-module with separate continuous module multiplication. Then B × A in the product topology is a unital Ql-algebra. Moreover, if B is a Q-algebra which is a commutative7 A-module with separate continuous module multiplication, then B × A in the product topology is an unital Q-algebra. Proof. Let first B be a Ql-algebra, then QinvlB is a neighborhood of zero in B. By the assumption, the map fa : B B, defined by fa(b) = ab for every a A and b B, is continuous. Therefore, there exists a neighborhood OB of zero in B such that f−a−1 (OB) QinvlB. Thus, −a−1b QinvlB for each a InvA and b OB. Hence, −a−1b has the left quasi-inverse (−a−1b)q −1 in B. Therefore, (b, a) with a InvA and b OB is left invertible in B × A and its left inverse is (−(−a−1b)q −1a−1,a−1). So8 OB×InvA Invl(B × A). As OB×InvA is a neighborhood of (θB,eA) in the product topology on B × A, then Invl(B × A) is also a neighborhood of (θB,eA) in the product topology on B × A. Consequently, B × A is a Ql-algebra. The proof is similar if B is a Q-algebra and a commutative A-module. Using the Proposition 2.1, we give a family of examples of Q-LFpg algebras. Let A be a unital locally pseudoconvex F -algebra, which is also a Q-algebra, and B a Q-LFpg-algebra, defined by a family of locally pseudoconvex F -algebras with α I. Then B = −→ Bα. 6A topological algebra is a Ql-algebra if the set of left quasi-invertible elements in A is open. 7That is ab = ba for each a A and b B. 8Here Invl(B × A) denotes the set of left invertible elements in B × A.
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