can i put a uv sterilizer in a 12 gallon nano cube?

do you have a problem in the tank? i would put it very low on the "must have" list. Do you have RO/DI or a protein skimmer? these are things that i would look into first of all.

UV's can be run on any tank. But the bigger question is why? They aren't needed for a healthy tank, and in fact, particularly in reef setups, reduce the good diversity. UV's sterilizers will kill the free-swimming plankton--corals, worms, snails, and most fish all spend time as free-swimming larva, so a UV will prevent them from ever maturing. Not good. And, especially with corals, since the corals consume plankton! Even for controlling pathogens, a good hospital tank will serve you better than a UV sterilizer, since you can treat more aggressively in a hospital setup than you can in a main tank, and UV's won't kill all the parasites that may be in a tank, just those that are pulled through it.

Sorry--I feel obligated to clarify this.

Having a skimmer is in no way related to the need for a UV sterilizer. They do completely different things. A skimmer removes dissolved organics from the water, and helps keep the water clean. UV skimmers kill anything living, free-swimming and floating in the water, such as plankton and algae. Killing them does not remove them from the system, it just kills them. Most tanks will benefit from a skimmer, very few will benefit from a sterilizer and some will in fact suffer from it, as the dead plankton can result in serious algae food.