While an air compressor heater and a pressure drop "vortex" cooler could be construed as a heat pump with some creative marketing and/or play on words, the system is not really considered a "standard" heat pump. As the others have stated, the compression/decompression of atmospheric air is really not energy efficient.
The missing pieces are the closed loop and the condensible gas. Even a machine with no moving parts except the compressor (think deep freezer chest) can cool (or heat, think water heater cylinder) a lot of mass. Latent heat transfer is the key.
Usually there's no such thing as a free lunch, but in this case, the engineers figured one out. Its called an EVAP chiller or a swamp cooler. Please don't make me explain how that works.
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