Arens regularity and local reflexivity principle for Banach algebras
Let E be a non necessarily associative Banach algebra. Using ultraproduct techniques, we define on \(E^{**}\) a product \(_{(J^ 0,{\mathcal U})}\) depending on some ultrafilter \({\mathcal U}\) and extending the original one on E. Though generally different from the Arens products, it coincides with them (and E is Arens regular) if it is separately \(w^ *\)-continuous. Automorphisms, antiautomorphisms and multipliers on E extend to \((E^{**},_{(J^ 0,{\mathcal U})})\). We give some criteria for the associativity of \(_{(J^ 0,{\mathcal U})}\). Finally we associate to each \(_{(J^ 0,{\mathcal U})}\) a topology on \(E^{**}\), and compare it with the Mackey topology.
scientific article; zbMATH DE number 4058200
scientific article