I would like to know how to use a battery to charge a piece of plastic like a feather duster.
I know static electricity is supposed to come with friction for example. But is there a way to do it with a battery to maintain a charge level in the air?
These are two related but distinct processes. There is nothing static about the electricity from a battery, charges are keep flowing. However, there is something to be said about capacitance, if you attach the battery at both ends, the feather will charge ever so slightly, and that charge will either be released into the air or onto the next thing that touches it depending on things like the humidity of the air. Though, this capacitance will basically be negligible and depending on your application, you would never notice it, unless you were using a highly capacitative material.
Static friction is caused by free electrons from one object "rubbing" off onto the other by as you said, friction etc. This creates a p.d across the duster because the positive charges in the duster move to one side, closer to the electrons on the cloth/object that is rubbing, and the free electrons in the duster move to the opposite side as they are repelled by the electrons from the cloth. This makes one side of the duster positive and the other negative. Compared to the capacitance, this is a huge amount of charge. If you are trying to make it last longer in air you should rub more. A battery will not help here. Otherwise, use a capacitor for a similar effect.