All of the buildings I’ve lived in (not counting dorms at college), are simply called by their numbers and street name. Swanky buildings have names. Really swanky buildings have names with definite articles like, The Ellington, or The Market Steps. If they’re ultimately swanky, they get to name the street or place where they are located and then call themselves something like One Liberty Plaza — with "One" written out.
My friend Lizard lives in what appears to be a great building. You can tell because it’s called The Centennial. I was thinking it would be a great place to live until I, like everyone who entered or exited the building that weekend, saw this in the building’s entrance:
Even with a definite article, I would never want to live in a building with managers who tempt and torture their tenants like this.