The AC does not let out cold air. The band seems to be missing.
If by the band you mean the belt, then yes, the reason you have no air conditioning is because the belt is missing. The belt is run over the crankshaft pulley, a tensioner to keep the belt tight and the air conditioning compressor. The belt could have cracked from old age and broker off or the tensioner pulley or the compressor pulley could have frozen and burned the belt off. A proper inspection should be carried out by a qualified technician, such as one from YourMechanic, in order to verify that all the pulleys in the system operate properly and to install a new belt.
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