I see what you are saying.
Your idea is that you use energy to liquefy a gas (O2 in your example), and then use ambient heat to expand it, doing work. Then, remove heat from expanded gas, and continue.
This really isn't anything new though; it is essentially a steam cycle except it uses a working fluid much more volatile than water, and uses ambient heat/solar power as a heating source instead of a more conventional fuel source (nuclear, coal, etc.).
It will NEVER be able to produce more work than is put in as energy, but there is no rule that says it can't produce more energy than YOU put in as work (in this case, the liquefaction process and pump required to keep the liquid flowing).
This is similar to how, from the point of view of work/energy put in, an oil rig produces huge quantities of energy. All you have to do is pump the crude oil up; the biochemical processes that guided its formation already happened, free of charge. This is similar to using the sun; an extant free energy source.
Your process MAY work, but the amount of work produced will not exceed the difference between the gas in its highest enthalphy state minus its lowest. (I think...correct me if I am wrong, anyone, but I believe this is technically accurate.)