A UV sterilizer might help with water clarity issues, but I don't think it will slow green hair algae down.
A UV sterilizer pumps water through a chamber where it is exposed to ultraviolet (UV) radiation. Strong doses of UV light play merry heck with DNA, which is part of the reason why tanning enthusiasts often get skin cancer. Anyway, UV also denatures many proteins. By breaking down proteins, a UV sterilizer can improve water clarity.
Many bacteria in the water column will be killed by a trip through the UV sterilizer. On the other hand, most sterilizers have a fairly low flow rate, so there's plenty of opportunity for normal bacterial growth rates to make up for the deaths caused by the UV.
Bob Fenner, among others, is a fan of UV sterilizers because they also increase the reduction/oxidation (redox) potential of the water. Other experts in the field say that the benefit of increased redox potential is offset by the fact that most American hobbyists don't have redox meters. "Don't add any supplement you can't measure" is a piece of advice that's been around for a while.
One of the ways hair algae can spread is when tiny fragments torn off of established filaments land somewhere in the tank and begin reproducing. A UV sterilizer could potentially kill some of those fragments while they were drifting in the water column. Unfortunately, hair algae also uses several other reproductive strategies, so having a UV sterilizer probably won't slow it down very much.
