Need dating advice... Can you be friends with someone you've dated in the past while in a new relationship?

For context, my bf continues to talk to this girl he briefly dated before us. He chose not to pursue a relationship with her and decided to just be friends despite her wanting more.

When we first started dating I went through his phone and found flirty text messages between them. That was over a year ago and he said he'd stop speaking to her. I asked him why he keeps her around and his response is always that he thought I was messing around with my gut best friend which I wasn't. He says they started communicating again after he had his suspension. Once he realized his assumptions were false he apologized and I said goodbye to my guy best friend because I didn't want to jeopardize my relationship.

With the year and a half we've been together I've found out on several occasions that he's still in communication with this girl. They're not flirting anymore but he's being sneaky and didn't admit to it until after I discovered the messages.

Should I end my relationship? To be honest what has kept me around so long is knowing they haven't been sexual. But is that enough to stay? I think it's emotional cheating but idk.