Professor Reynolds has some questions about the legal status of space elevators.
Here are some more. The problems associated with anchoring such a beast in an unstable and/or corrupt equatorial country has caused many of those planning such things to put them instead on floating ocean platforms, in international waters. This raises some new issues, because now, instead of (as Glenn notes) the structure simply being a very high tower, it would be a tall ship that would put to shame all of the previous false claimants to that designation, with their puny little sticks for masts. This thing would have a crows nest above geosynchronous altitude. It would also imbue an entirely new meaning to the old phrase, the “high seas.”
Presumably, maritime law would apply to it, absent any codicils to the Outer Space Treaty. Thus gambling would be legal, the person in charge of the facility would be captain of the ship with all implied powers (including wedding officiation), etc. He would also keep the keys to the gun locker.
As to whether such an object would be “launched” or not, that depends on the construction technique. Most plans for these things would in fact involve launching them, in the conventional usage of the term. It would be only after the initial mass had been deployed at geostationary orbit that it would be slowly lowered and attached to the planet. After that, more mass would be added via the elevator route, so it would really be a hybrid creature, partially launched and partially constructed from the surface.
It would get even more interesting if the orbital anchor were a captured carbonaceous asteroid, and the diamondoid materials needed for the tether manufactured from it. In that case, it would be more akin to a grappling of the earth from another planetary body. I’m not sure what the legal implications of that would be. Would its status somehow change once it was actually attached to the planet? And what does “attached” mean, if the bottom of it is simply floating on the ocean or, even more tenuously, in the atmosphere?
This is one that will indeed keep the lawyers happily busy for years.
[Update at 8:30 AM PDT]
This strikes me as a loophole in the sovereignty restrictions of the OST. If an asteroid were the orbital anchor for the beanstalk, and it were designated a ship (flagged, perhaps, in Panama or Liberia), then the property rights to mining (and owning) it would presumably be settled. Or at least much more so than they would be for it as a freely orbiting body. And one could protect additional objects in like manner by simply attaching them.