You've figured out one half of the Hall-Heroult process. Just melting alumina on its own won't liberate the oxygen. The reason is difficult to explain without getting into chemical thermodynamics - one way to think about it is that the lowest energy state is one where oxygen is bound to alumina - where the electrons from alumina have been "taken" by the oxygen. In diatomic oxygen, the electrons are "shared" (the bonds are covalent) and so the total energy is higher. This is a very brief and not-very-detailed explanation, but the end result is that oxygen and aluminum, if given the opportunity, would rather form a bond than not.
When you add thermal energy, unless it changes that balance (it doesn't, at least not until you get way past the melting point), all you are doing is making it more possible for the reaction to happen. If you kept going, you could eventually add enough thermal energy to completely dissociate the alumina, at which point (depending on how you did it) you might be able to remove the oxygen and recover the aluminum.
However - the amount of energy it would take to do that is extremely large. Instead, it turns out to be much easier to simply force oxygen to give its electrons back to aluminum using an electric potential. When you do that, you electrolyze the compound and create diatomic oxygen at the anode and aluminum at the cathode.
The Hall-Heroult process takes this another step to make this work at an even lower temperature (1000 C) by dissolving the alumina in molten cryolite. As a result, it is one of the most (maybe the most) efficient industrial process for producing aluminum, and is responsible for the widespread availability and low cost of aluminum in modern times (and also for the existence of Alcoa).
Your idea about using solar energy isn't bad - although you might be surprised at the amount of energy you need to produce aluminum - both to melt it, and to electrolyze it. One thing you might look into is using solar thermal heating (with a collector, for example) to melt the alumina/cryolite mixture, and then use photovoltaics just for the electrolysis step. That would be quite a bit cheaper than using photovoltaic electricity alone to run an electric heater.
Industrially, the cryolite mixture is kept molten by passing a large electric current through it. The electricity is usually produced by power plants, since the amount required is so large (15 kWh/kg according to Alcoa, and that's just for the electrolysis step!). Although coal and gas power isn't renewable, the advantages are a relatively low cost per unit of power (which might not stay true in the next few decades) and on-demand power generation (likely to always be a big advantage over solar).
There is also nuclear, hydroelectric, and geothermal power - all of which provide large amounts of on-demand power very inexpensively, once the plants are built.