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).
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