Inductive limits of locally m-convex algebras

Abstract

In this short note we prove that the locally convex inductive limit of commutative locally m-convex algebras is already locally m-convex whenever multiplication is continuous. A topological algebra is an (associative) algebra endowed with a vector space topology such that multiplication is jointly continuous. If one considers inductive limits (in the category of topological vector spaces) of topological algebras multiplication may fail to be continuous since it is a bilinear map on the product which is certainly not linear, and forming inductive limits only respects the linear structure (although this is quite clear, the dierence between linearity and bilinearity in this context has been overlooked several times in the literature). The most important category of topological algebras is that of locally m-convex (l.m.c.) algebras having a 0-neighbourhood basis consisting of m-convex sets (i.e. absolutely convex sets B which are multiplicative in the sense that B 2 B). This class was introduced by Michael [9] and Arens [2] since it allows generalizations and applications of the theory of Banach algebras. Since their work a number of authors studied the question whether the inductive limit (in the category of locally convex spaces) of l.m.c. algebras is again l.m.c., see e.g. [1, 3, 4, 5, 7, 8, 10]. This is indeed true for countable inductive limits of normed algebras [1, 4]. On the other hand, Warner [10] gave an example of an inductive limit of metrizable l.m.c. algebras which is not l.m.c., and in [4] it is shown that [