The Peacemaker- Or Not?

Question: how involved do I want to be, as a manager, in the lives and/or conflicts of my co-workers?
The thing is, I am about to go to my evening shift with Jane and some other weird guy. Last time I saw Jane was the day she made Mallory cry for whatever reason (whatever Sanjay-related reason, that is).
I myself am worried, just because I'm starting to fear her general wrath, that she's going to make me cry tonight. When I'd first learned of her and Mallory's beef, I wanted to intervene. In the past, when people have been duking it out at work, I've stepped in just to calm things down and at least get them to cool off until after work.
At this moment, my plan is to go into work, be as cool and charming as possible (as I always am) and hope that nothing gets brought up through casual conversation that I don't want to talk about. I will avoid at all costs the subject of Jane being a bitch, and I will hope this saves me from possibly being bitched at.
I feel like it's the right thing to do- to stay out of it and mind my own business... for now.
I've decided that unless people are messing with eachother on one of my shifts, then I should let them live their lives. Am I going about this correctly? It's not like I'm the human relations department at work or anything- I'm just a warm-hearted, caring human being...ummm...yeah.