I am in the process of buying a small business that is currently bank owned. I will establish a lease with the bank. If the bank sells the property do I have to re-negociate a lease with the new owner or does the lease I have carry over until the expiration date?

I'm not following. Are you planning on buying the property or just renting it from the bank?

I am buying a business that is in a mall that bank owned and would be leasing the building that the business would be conducted out of that building.

So, you plan on buying the property, and then subletting it?

If the bank owns the building at the time you sign the lease, and if the lease doesn't say anything about the bank having the right to terminate the lease upon a sale of the building, then the new owner will have to honor your existing lease.