Is it possible to use refrigeration technology to cool down the aircraft's skin so it can reach much higher speeds?
Refrigeration technology moves heat energy around, it doesn't get rid of it. Your fridge moves heat from inside to the outside.
To get rid of it, you have two means, broadly speaking:
- Heat up some mass and throw it away
- Radiate it away as light/IR
Air cooling fundamentally heats up the surrounding air, and moves the now hot air away. I believe the SR-71 used it's fuel as heat sink as well, heating the fuel before combusting it, and ultimately throwing it away.
Radiating it is inefficient at low temperatures. And it's directly dependent on surface area, which is at a premium on a plane.
So in essence you'd have nowhere to put the heat energy you removed from the skin of the aircraft...