4 MATI ABEL AND REYNA MAR´ IA PEREZ-TISCARE˜´ NO A topological algebra (E, τ) is a Q-algebra if the set QinvE of quasi-invertible elements2 (in case when E is a unital algebra, then the set InvE of invertible elements) of E is open in the topology τ. We shall say that an LFpg-algebra (LFp-algebra) (E, τ) is a Q-LFpg-algebra (respectively, Q-LFp-algebra) if (E, τ) and all algebras (Eα,τα) in the inductive limit are Q-algebras. Similar topological algebras for locally m-pseudoconvex case3 have been considered in [10] and for locally m-convex case in [5] and [11]. Examples of Q-LFpg-algebras and Q-LFp-algebras are given in the present pa- per. It is shown that (E, τ) is a Q-algebra if it is an inductive limit of Q-algebras and τ is the inductive limit topology on E. Conditions, in order a locally pseudoconvex algebra (E, τ) (which is an inductive limit of locally pseudo- convex Q-algebras and τ coincides with the locally pseudoconvex inductive limit topology on E, defined by canonical maps) is also a Q-algebra are found. Moreover, conditions in order that LFpg-algebra (LFp-algebra) is a Q-LFpg-algebra (respectively, a Q-LFp-algebra) are given. 2. Q-LFpg-algebra and Q-LFg-algebras First we give some examples of Q-LFpg-algebras and Q-LFg-algebras. 2.1. Examples of Q-LFpg-algebras and Q-LFg-algebras. Let (A, τ) be a unital commutative locally pseudoconvex F -algebra with continuous inversion which is also a Q-algebra, where τ is defined by kn-homogeneous seminorms pn with kn (0, 1] for each n N, and let K(R,A) be the algebra of continuous functions from R to A with compact support, in which the algebraic operations are defined by (f + g)(x) = f(x) + g(x), (λf)(x) = λ(f(x)) and (fg)(x) = f(x)g(x) for each f, g K(R,A), λ K and x X. Moreover, for any integer i 1 let K(R,A [−i, i]) be the subalgebras of K(R,A), which support of every function is contained in [−i, i]. The topology τi on K(R,A [−i, i]) for each i is defined by kn-homogeneous seminorms qi,n, where qi,n(f) = sup x∈[−i,i] pn(f(x)) for each n N and f K(R,A [−i, i]). Then (K(R,A [−i, i]),τi) is a metrizable locally pseudoconvex algebra for every i. To show that (K(R,A [−i, i]),τi) is com- plete, let i N be fixed and (fk) be a Cauchy sequence in (K(R,A [−i, i]),τi). Then for every ε 0 and n N there exists N 0 such that qi,n(fk fm) = sup x∈[−i,i] pn(fk(x) fm(x)) ε whenever m k N. Hence, pn(fk(x) fm(x)) ε for every x [−i, i] (also for every x R, because the support of function fk is contained in [−i, i] for each k N) and every n N whenever m k N. It means that for every x R 2An element a is quasi-invertible in an algebra A, if there exists another element b A such that a + b = ab. 3A topological algebra A is locally m-pseudoconvex (locally m-convex), if it has a base of neighborhoods of zero which consists of absolutely pseudoconvex (respectively, absolutely convex) and idempotent sets U (that is, UU U).
Previous Page Next Page