To make this transition as easy as possible for both you and your former peers, identify which "crucial conversation" you can engage them in. Doing so will help remove unnecessary awkwardness and tension and help both of you to adjust to your new role and relationship.
1. Acknowledge the "Elephant in the living room". - Let them know you can imagine it might be awkward for them and - if it is for you, too - acknowledge this.
E.g. "Sam, I want to check in with you about how things are going with me being a supervisor after having been a co-worker. I can imagine if the roles were reversed, I might feel a little strange about that and wanted to see how it's going for you..."
2. Ask them for their thoughts on how you and they can make this transition as smooth as possible for everyone.
E.g. "Is there anything I can be do to make this smoother and is there anything I could make sure I NOT do?"
3. Avoid preaching the obvious - e.g. "We all have a job to do and I need to count on you... blah, blah, blah..." Any officious, formal, Old School boss lectures are guaranteed to alienate.
4. Ask them for input on what you can do to be the best possible supervisor for them. Just the fact you care enough to ask makes a huge difference.
E.g. "Brenna, I obviously want to do the best job possible at this, which means doing the things that help you be your best and not get in the way of that. I know I just got started in this position, but is there anything you see that I could be doing more of, or less of, that would be good...in your opinion?"
5. If a peer also applied for the job, depending on your relationship with them, you might ask them how they are doing with your getting the job. If you don't have a particularly close relationship with that person, you might choose to simply acknowledge in a low key way that if they want to say anything about how they're doing with this, you're open to hearing about it. If you don't believe they would be comfortable talking about this at all, then you can stick with the above ideas.
E.g. "Chloe, I know you applied for this job too and I wanted to check in with you about that and how you're doing with me getting it and being the supervisor now." If they acknowledge some unhappiness or awkwardness, you can ask something like "Is there anything I can do to help make it more comfortable?"
6. If a peer who you've been close to appears to be taking advantage of that friendship, address this quickly. You can do this without sounding overly formal or officious.
E.g. "At the risk of stating the obvious Sam, the fact that we're friends can't get in the way of you doing your job and being held accountable like the others. Maybe I'm reading into things too much, but some of the things that have been going on, like _______ and _______ make me wonder about this... What's your take on this?"
While nothing works 100% of the time, having the above conversations will help make your transition from peer to supervisor smoother, easier, and faster.
From Peer to Supervisor: 6 Tips for an Easier Transition