Properties ofTQ-algebras

Abstract

Several properties of unital left (right) T Q-algebras are described. The conditions when a unital semitopological algebra is a left (right) T Q-algebra are given. It is shown that the space M(A) (of nontrivial continuous multiplicative linear functionals on A) in the Gelfand topology is a compact Hausdorff space for every unital T Q-algebra with a nonempty set M(A) and a commutative complete metrizable unital algebra is a T Q-algebra if and only if all maximal topological ideals of A are closed. Examples of T Q-algebras are given. Open problems are presented. Here and later on every U denotes the closure of U in A.2 That is, the net (a λ ) λ ∈Λ is the same in (a λ a) λ ∈Λ and (aa λ ) λ ∈Λ . 3 It is known (see [2], Corollary 2) that every complete unital locally m-pseudoconvex algebra is an invertive algebra, but every commutative complete metrizable unital algebra with a discontinuous inverse is not (see [32], Proposition 4).