Why doesn't water undergo electrolysis?
Why does Na2SO4 undergo electrolysis?
This may be a question of semantics.
Water does electrolyze.
Na2SO4 does not.
BUT, water has such a low ionization that it doesn't conduct electricity very well (due to the lack of ions). Adding a little Na2SO4 provides the ions to conduct the electric current; under those conditions H2O is electrolyzed to H2 and O2.