When you take on your new role, team dynamics are going to change.  Don’t let past relationship issues come into play; start fresh with each employee.  Talk with them individually, establish that you expect a fresh start and give everyone an equal chance to do a good job for you.  This is not easy to do, especially if you were peers with an individual that you never got along with, or you competed with them for the promotion.  You are the boss now and must take the high road.  Be humble, but don't let the issue between you get ignored by saying nothing.  Be strait forward, friendly and clear. An example is: "I know we have never really gotten along and I can't say that I ever had a good reason for it.  I want you to know that what ever the reason, it is in the past and I want this new relationship to be a positive one.  I want you to be successful, and I want our team to be successful.  I am fully on board to make that happen and I would like a commitment from you to do the same."  If they can't commit to supporting you right away, give it a little time.  Be consistent with your approach and understand it may take a little time to get their buy-in.  However after three months if they are not supporting you and pulling their weight on the team, or if they have a bad attitude...take them aside in a formal sit down and give them a warning.  Three strikes and you need to let them go.  Your intention however must be the success of your team, nothing else.