Algebras of operators as topological algebras

Abstract

An example of a non-topologizable algebra was given in [2]. In [4] Żelazko gave a simple proof of the fact that, if X is an infinite-dimensional vector space, then the algebra of all finite-rank linear operators on X is not topologizable as a topological algebra. In the following we use a similar idea to prove that, if E is a Fréchet space which is not normable, then each subalgebra A of the algebra of all bounded linear operators on E such that A contains the ideal of continuous, finite-rank operators, is non-topologizable as a topological algebra. This is a shorter proof and more general version of the result of [1].