I've wondered about this too. Since most aquariums are set up to minimize algae, it doesn't live long in them. You could set up a tank to favor phytoplankton, eliminating all mechanical filtration, skimming, etc. and keep nutrient levels higher, so it could flourish. If you are successful, you'll likely end up with a tank full of green water.
It would great to find a happy medium, so it could live indefinitely, without clouding up the water. Tough to do in a low nutrient reef tank, but maybe in a high nutrient planted tank (like mine) it could work. I may give it a try at some point. If so, I'll be sure to report back to RC.