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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