Should I be ok with my boyfriend messaging girl he used to like if its innocent?

Anonymous
Been going out with my boyfriend for a while now (bout 21 months) and its generally a pretty good relationship. A while back I saw some messages pop up from a girl he used to work with, this happened in the early hours of the morning whilst we were in bed together, naturally I was fuming. He assured me it was just because she was out and had seen a colleague so wanted to tell him about it and said nothing at all was going on, nothing had ever gone on with them etc. he said they sometimes messaged a bit when drunk but nothing flirty or anything, just catching up. I trust him so dropped it and moved on. Anyway, a few nights ago he was drunk and accidentally mentioned that she'd messaged him recently saying that she had a boyfriend now, and he let slip that he had actually asked her out a few times before he met me. Am I wrong to not like him talking to women that he used to like, even if he does say it is innocent? Its happened a few times where he's got messages from people he used to date or whatever, is that normal? he's promised me that he won't talk to any women from now on, should I trust him?
Should I be ok with my boyfriend messaging girl he used to like if its innocent?
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