Characterisations of von Neumann algebras

We give various characterisations of W⁎-algebras as C⁎-algebras possessing certain locally convex topologies. First, we prove some (related) algebraic characterisations that give, among other things, a strengthening of a result of Kadison and a consequence that a C⁎-algebra is a W⁎-algebra if (and only if) it has a (priori not required to be isometric) predual with respect to which the Jordan product is separately weak⁎-continuous. Based on these algebraic characterisations, simple tricks will then give us a topological characterisation of W⁎-algebras among the class of C⁎-algebras that improves the statement of Sakai's theorem (and incidentally gives a new, simple proof of this classical result). Finally, as a variation of this characterisation, we prove that W⁎-algebras are those C⁎-algebras that admit a locally convex topology τ weaker than the norm topology such that the closed unit ball of every maximal commutative C⁎-subalgebra is τ-compact. This then gives us a strengthening of Pedersen's well-known characterisation of W⁎-algebras among the class of AW⁎-algebras.