can black itch live in your tank without host

You have to run your display tank fallow (fish less) for about 2 months to get rid of ich. You must also take action with the fish you have in QT by either copper dosing or hyposalinity to kill the parasite. Then any future fish you purchase must in QT and treated or you risk reintroducing ich back into your system.
+1 salt

On the other hand, a tank contaminated with ich is not the be-all-end-all. Many people have ich in their tanks, but if you keep your parameters stable, feed high quality food, and make your tank as stress-free as possible, your fish can fight of ich easily.
It can survive in a fishless system for several months. If you really want to get it out of your display tank, you need to pull all the fish out and let it run fallow as mentioned above. After that, you have to quarantine all new arrivals and keep LFS water out of your system.