Can I deny a customer from redeeming a gift card that was sold by the previous management?

You say the previous owner signed the gift certificate , presumably at the time when the previous owner owned the business.  If the business issuing the gift certificate is the same entity and just has a new owner (i.e. if the business is xyz corporation and you bought all the stock of xyz corporation), it is still obligated on the gift certificate.  If the business is now a completely separate entity which merely bought the assets of the previous owner (i.e. the previous business was xyz corporation but abc corporation bought its inventory, goodwill, receivables, etc.) and didn't assume any of its liabilities, then it may not be responsible for the gift certificate.  However, whether technically liable or not, I would think that the damage to goodwill from not honoring a gift certificate would be a lot higher than the cost of the goods bought with the card.