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There's two different things that could be going on here.
She met someone or dated someone who hurt her really badly, and she's got cold feet about dating. In which case, you say hello and be friendly at the public events, and if and when she's ready to show interest in you, play it by ear, and play it easy - or she'll run.
The other option like others have said - she's not into you and is trying to be nice about it. It could be that she's really not looking for a relationship. Just keep things friendly when you see her, pushing her will make her run far and fast.
I think you like her and are trying to talk yourself into her liking you. She doesnt'. If a person likes someone they would eventually respond. A lunch is very low stress so it should be no problem. You may have come on strong too. Sounds like you really like her.
Some people can't seem to accept that a person just isn't interested in them - instead, they have to try to analyze the person and try to figure out what's REALLY going on, when the only thing that is going on is lack of interest.
It's a male ego destroyer when women who are not really interested in them make excuses on why she didn't fall over backwards to be with you...or the other guy.
She has her choices and she is going to make them without you..she is NOT DESPERATE