Rob Yeung's Answer:
Disrespect is an interesting concept. Because disrespect occurs purely in the eye of the beholder. It doesn't matter what you have or have not done. If people feel disrespected, that is their reality – that's the way they feel.
So this older woman may not like you because she may feel aggrieved in some way. Perhaps she feels that she has been disrespected because you got promoted over her – maybe she feels she has better skills or experience. Or maybe there is some other reason.
However, you can't know the reason unless you ask her why. The last thing you may want to do is to speak to her, but I'm afraid that is probably the best thing for this situation.
Arrange to meet with her privately, one-on-one in a quiet location. She may resist the idea of spending time with you, so be persistent and patient but do keep asking until she accepts your offer to meet for a chat. Ideally, try to take her out of the office – perhaps for a sandwich lunch, a coffee, or a drink after work somewhere quiet and away from the prying eyes and ears of colleagues.
When you meet, tell her that you're trying to do the best as a new manager but that you feel that she isn't entirely happy with you. Use "I feel" statements (e.g. "I feel that you aren't happy with me") rather than "you" statements (e.g. "you have been stirring things up") because the former sound far less accusatory. Then invite her response. Allow her to vent her frustrations.
It's difficult to predict what she may say. Some people feel better just for the fact that they have had the discussion. Others may feel so hurt by some imagined slight that they may never warm to you. But the first step is certainly to have the open conversation and find out the issues. You can only deal with what you know about.
Open dialogue and a big dose of empathy is a great start whatever the issues. I wish you the best of luck.