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 $|\cdot|_M$ on $A$, associated with $M$. We show that the closure of $M$ with respect to $|\cdot|_M$-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 $|\cdot|_M$-topology. Then we omit the Archimedean condition and show that there still exists a lmc topology associated to $M$, pursuing the same properties.