If this is correct, the sense development would be from "loft, ceiling" to "sky, air." Buck suggests a further connection with Old High German louft "bark," louba "roof, attic," etc., with development from "bark" to "roof made of bark" to "ceiling," though this did not directly inform the meaning "air, sky" (compare lodge (n.)). But Watkins says this is "probably a separate Germanic root." Meaning "gallery in a church" first attested c. 1500. From 1520s as "apartment over a stable" used for hay storage, etc.
The word once had a twin, Middle English lift "the air, the atmosphere; the sky, the firmament," from Old English lyft "air" (see loft (n.)).
Intransitive sense of "to rise, to seem to rise" (of clouds, fogs, etc.) is from 1834. Figurative sense of "to encourage" (with up) is mid-15c. The meaning "steal, take up dishonestly" (as in shoplift) is 1520s. Surgical sense of "to raise" (a person's face) is from 1921. Middle English also had a verb liften (c. 1400). Related: Lifted; lifting.
c. 1300, "invisible gases that surround the earth," from Old French air "atmosphere, breeze, weather" (12c.), from Latin aer "air, lower atmosphere, sky," from Greek aēr (genitive aeros) "mist, haze, clouds," later "atmosphere" (perhaps related to aenai "to blow, breathe"), which is of unknown origin. It is possibly from a PIE *awer- and thus related to aeirein "to raise" and arteria "windpipe, artery" (see aorta) on notion of "lifting, suspended, that which rises," but this has phonetic difficulties.
In Homer mostly "thick air, mist;" later "air" as one of the four elements. Words for "air" in Indo-European languages tend to be associated with wind, brightness, sky. In English, air replaced native lyft, luft (see loft (n.)). In old chemistry, air (with a qualifying adjective) was used of any gas.
To be in the air "in general awareness" is from 1875; up in the air "uncertain, doubtful" is from 1752. To build castles in the air "entertain visionary schemes that have no practical foundation" is from 1590s (in 17c. English had airmonger "one preoccupied with visionary projects"). Broadcasting sense (as in on the air, airplay) first recorded 1927. To give (someone) the air "dismiss" is from 1900. Air pollution is attested by 1870. Air guitar is by 1983. Air traffic controller is from 1956.