Ignoring topological structures of von Neumann algebras, the larger category of Baer $*$-rings was emerged. In the unique text written by Sterling K. Berberian (1), some concepts and results in von Neumann algebras are algebraically given so that they extend to Baer *-rings.
Recently, in two successful attempts (2), fundamental decomposition theorems of Wold, Halmos-Wallen and, Nagy-Fiaos-Langer have been algebraically proved so that one may conclude them in Baer *-rings.
Q. List any fundamental result in von Neumann algebras that can be stated algebraically. They can be reassessed to see whether they can proved algebraically!
Example. Putnam-Fuglede theorem.