On the topologies induced by a cone

Abstract

Let $A$ be a commutative and unital $\mathbb{R}$-algebra, and $M$ be an Archimedean quadratic module of $A$. We define a submultiplicative seminorm $\Vert \cdot {\Vert }_M$ on $A$, associated with $M$. We show that the closure of $M$ with respect to $\Vert \cdot {\Vert }_M$-topology is equal to the closure of $M$ with respect to the finest locally convex topology on $A$. We also compute the closure of any cone in $\Vert \cdot {\Vert }_M$-topology. Then we omit the Archimedean condition and show that there still exists a lmc topology associated to $M$, pursuing the same properties.