Luc/MathFox/Dan and many others -- thank you for a fun discussion.
Note as to reasons why we use a compressor, per the original question.
Chris: Why do we compress - to release more energy to do useful work per unit time within the engine (be it piston or gas turbine).
Yes.
Slippery_Pete :It's so that less heat is added to the air over the entire cycle.
Not quite.
That may be an outcome of the process of burning oxygen to release energy in the fluid to get something to turn (shaft or turbine) but it isn't the reason to do it.
Other point made earlier: without a delta P, (Pressure differential between inside the turbine engine and the outside air) you get no work out of the turbine, be it in turbo shaft, turbo prop, turbofan, or turbojet.
You build turbine engines to get work out of them.
My degree isn't important, but I did take some thermogoddamics and a few courses on compressible flow and turbomachinery. Otto, Brayton, Carnot ... all familiar names.
Put energy in, get directed energy out to make something turn.
That is why you have an engine.