On algebras with locally convex topologies admitting Arens products in their second topological duals

Abstract

We study algebras with locally convex topologies admitting Arens products in their second topological duals. We identify the exact conditions required for the existence of Arens products. We show that for algebras admitting Arens products in their second duals, the identity $WAP\left(A\right)=A^{⁎}$ is equivalent to Arens regularity (Pym's criterion). We show that on any infinite dimensional normed algebra $(A,\Vert \cdot \Vert )$, there exist uncountably many locally convex topologies τ compatible with the duality $〈A,A^{⁎}〉$, such that $(A,\tau )$ admits Arens products in its second topological dual. If $(A,\Vert \cdot \Vert )$ is Arens regular, strongly Arens irregular or extremely non-Arens regular, then there are uncountably many locally convex topologies τ on A for which $(A,\tau )$ has the same property.