Can you keep the "Full Name" accross Site Collections?

Is it possible to keep the "Full Name" accross site collections.

If I create an FBA user and give them access to a second site collection, their name appears as i:0#.f|fbamembershipprovider|sclarke instead of "Susan Clarke" in the second site collection.




I haven't done it, but you should be able to by setting up profile synchronization: