A store is private property like any other. Commercial establishments are open for commerical purposes. But, without knowing the facts behind your one line posting, it is true that the owner of private property has the right to exclude entry to those who they wish to exclude.
This answer is offered for informational purposes only. It is not offered as, and does not constitute, legal advice. Laws vary widely from state to state. You should rely only on the advice given to you during a personal consultation by a local attorney who is thoroughly familiar with state laws and the area of practice in which your concern lies.
If store security told you that you are banned, then you can be arrested if you return. They are considered to be a representative of the store. And, the ban does not need to be in writing.
If you disagree with the ban, then you should write a letter to the manager explaining the situation and get his written agreement that you can come to the store (and carry it with you when you enter.)