Contemporary Mathematics
Volume 657, 2016
http://dx.doi.org/10.1090/conm/657/13087
Locally pseudoconvex inductive limit of locally
pseudoconvex Q-algebras
Mati Abel and Reyna Mar´ ıa P´ erez-Tiscare˜no
Abstract. A LFpg-algebra (LFp-algebra) E was defined in Abel and P´erez-
Tiscare˜ no (2013) as a locally pseudoconvex inductive limit (respectively, a
locally pseudoconvex inductive limit of an increasing sequence) of locally pseu-
doconvex F -algebras, which satisfies certain conditions. The case when every
locally pseudoconvex F -algebra in the inductive limit E is a Q-algebra is con-
sidered in the present paper. Conditions for E to be also a Q-algebra are found
and examples of such inductive limits of topological algebras are given.
1. Introduction
1. Let E be a unital topological algebra over K (the field of real numbers R or
complex numbers C) with separately continuous multiplication (in short, a topolog-
ical algebra). If the underlying linear topological space of E is locally pseudoconvex
(see [8], [9], [13], [14], [16] or [17]), then E is called a locally pseudoconvex algebra.
In this case E has a base U = {Uλ : λ ∈ Λ} of neighborhoods of zero consisting of
balanced (μUλ ⊂ Uλ, when | μ | 1) and pseudoconvex sets (Uλ + Uλ ⊂ μUλ for a
μ 2). This base defines a set of numbers {kλ : λ ∈ Λ} in (0, 1] such that
Uλ + Uλ ⊂ 2
1
kλ
Uλ
and
Γkλ (Uλ) ⊂ 2
1
kλ
Uλ
for each λ ∈ Λ, where
Γk(U) =
=
n
ν=1
μνuν : n ∈ N,u1, · · ·,un ∈ U and μ1, · · ·,μn ∈ K with
n
ν=1
| μν
|k
1
is the absolutely k-convex hull of a subset U ⊂ E for k ∈ (0, 1]. Herewith, a
subset U ⊂ E is absolutely k-convex, if U = Γk(U), and absolutely pseudoconvex, if
2010 Mathematics Subject Classification. Primary 46H05; Secondary 46H20.
Key words and phrases. Topological algebras, locally pseudoconvex algebras, F -algebra,
Q-algebras, locally pseudoconvex inductive limits of locally pseudoconvex algebras, LFp-algebras
and LFpg-algebras.
Research is in part supported by the Estonian Targeted Financing Project SF0180039s08 and
by the European Union through the European Social Fund (MOBILITAS grant No. MJD247).
c 2016 American Mathematical Society
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