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).