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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 1