All depends on the terms of the sale. You are assuming that ownership of the business entity was transferred. That is one option for a transaction, but one that comes with considerable risk as it entails assumption of all prior liability.

The alternative is to purchase the assets (including trademarks) of a business, assume the lease, and get a new business license. In this manner you effectively end up with the same business, but an entirely new legal entity. 

The latter case is far more common with small businesses as many small business are initially established in a manner that makes transfer of ownership hard, if not impossible.