Yeah, IIRC every planet has it's own local police force. The Arbites serve as the "Supercops". They serve two purposes, they keep an eye on the local government on behalf of sector government or the High Council, and they handle crimes of a more-than-local scope.

Each world/society can have it's own name for these local police. One of the Cain novels mentions that, rather than keep track of whether the locals call them "Marshals" or "Enforcers" or "Lawmen" or whatever, experienced interplanetary travelers just call them all "Arbites".

As I understand it, a murderer gets handled locally. If you murder somebody, then flee to another planet, then you're probably in the clear. If you murder somebody really important, then flee to another planet, then the Arbites will try to track you down, or they'll tell the local cops to track you down and hand you over to them.