For example, most performers on the Burlesque scene have accounts specifically for their Burlesque personas, with names like "Kitty Cupcake" or "Tabitha Taboo" and the like. These are obviously not their real names, but a consequence of the fact that Burlesque performers don't want the offline attention on their personal or professional lives outside the performing scene, and that it is standard on that scene to have an alternate identity, in the real world, who is the Burlesque artist.
The two identities are cleanly separated in real life as well as online, and they tend to use Facebook online to network with each other, get gigs, post photos, accrue fans, and so on. I know several such artists who can't be bothered to update their personal Facebook profile much, but are very active Facebook users under their Burlesque pseudonym, with many friends/fans/connections.
To consider these accounts "fake" seems, to me, like a mistake. These are very legitimate use of a social networking platform. It would be a mistake for Facebook to target such accounts and close them down. They would generate immense bad will amongst an increasingly influential community of artists.