Abstract

The main purpose of this paper is to investigate some topological properties of the double multiplier algebra on a topological algebra. Let Md(A) be the double multiplier algebra on a topological algebra A, and let u and s be the uniform and strong operator topologies on Md(A), respectively. It is shown, under some additional hypotheses on A, that(1) Md(A) is u- and s-complete;(2) A is a u-closed two-sided ideal in Md(A);(3) A is s-dense in Md(A);(4) s and u have the same bounded sets;(5) each continuous onto homomorphism ϕ:A→B has a unique extension ϕ˜:(Md(A),sA)→(Md(B),sB).