Ok, I have 2 mins to have a go.
Air supposedly the same temp after compression as before. Ok, I accept that. However, friction heating is occuring to the compressor blades whilst spinning, so new inlet air is not only heated by compression but also by the hot blades themselves of the compressor (in so doing cooling them). Since a tempraise of 20 deg c is relatively small compared to the overall temp rise from inlet to turbine inlet (prob the heat of the bleed air mixed for cabin heating), this effect could account for it. So in summary:
Air in theory should be the same temp before and after compression, PVT..... So additional energy must have been added, in this case (by my poor stab at reconning) blade friction heating/cooling.
Mr L.