Interesting post. My dad (who would be 106 if he were alive) was probably a young man when that was being built, and worked nearby. For reasons unknown, he had lifelong nightmares that "the Chrysler building was falling down."
I don't think he would ever have thought those dreams were a prescient if not allegorical forecast of this nation's economic state. I'm starting to wonder. :-
Seriously, as long as money can freely flow across borders, these things are going to happen. I don't have a strong conviction on whether or not this is a good or bad thing. It could just mean that others think we are still a good place to invest and therefore some of the money which routinely flows out of the US is coming back in.
Were this building a strategic resource, e.g., a railroad, I'd have grave concerns about foreign ownership.