Induced topologies on certain Banach algebras

Abstract

Let $\mathcal{A}$ be a Banach algebra with a bounded approximate identity bounded by 1. Two new topologies $\tau_{so}$ and $\tau_{wo}$ are introduced on $\mathcal{A}$. We study these topologies and compare them with each other and with the norm topology. The properties of $\tau_{so}$ and $\tau_{wo}$ are then studied further and we pay attention to the group algebra $L^1(G)$ of a locally compact group $G$.  Various necessary and sufficient conditions are found for a locally compact group $G$ to be finite.