Example of an Algebra Which is Nontopologizable as a Locally Convex Topological Algebra

Abstract

Let X be a real or complex linear space and denote by L(X) the algebra of all its endomorphisms. We prove that L(X) is topologizable as a locally convex topological algebra (with jointly continuous multiplication) if and only if it is topologizable as a Banach algebra and this holds if and only if X is of finite dimension. A topological algebra is a Hausdorff topological linear space equipped with a jointly continuous associative multiplication. A topological algebra is said to be locally convex if its underlying topological linear space is locally convex. The topology of a locally convex algebra can be given by means of a family (lixlll,) of seminorms such that, for each index a, there is an index fi such that (1) lIXyll(i ' !!XlflYl8yfl for all x and y in the algebra in question (see [4]). For general information on topological algebras the reader is referred to [1, 2, 3, 4] and the references therein. Note that some authors define topological algebras as topological linear spaces equipped with a separately continuous multiplication (cf. [2]). In [5] and [6] we posed the following question (Problem 2): Is it true that for every real or complex algebra there is a topology making it a locally convex algebra? In [6] we have shown that the answer is positive if we replace the requirement of joint continuity of multiplication by the weaker assumption of its separate continuity. We also conjectured (cf. [6, Problem 2a]) that the answer to the problem is negative if we take, as the algebra in question, the algebra of all endomorphisms of an infinite-dimensional real or complex linear space. The aim of this paper is to prove this conjecture, which gives the nontrivial implication in the result formulated in the abstract. Theorem. Let X be a real or complex infinite-dimensional vector space. Then there is no topology on the algebra L(X) of all endomorphisms of X making it a locally convex algebra. Received by the editors April 13, 1989 and, in revised form, August 24, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46H05.