Boyfriends ex won't leave him alone?

My boyfriend was with his ex for 4 years. In the end of the relationship she stole all of his stuff, maxed all his cards, and wiped his bank accounts. Then after they broke up she would call him to hang out, use him for sex for like a week and then stop talking to him. She would do that every few months. But he's finally been able to get over now that we are together. A few times she has sent him messages on Facebook of funny things she finds. Which that's not what I really have a problem him. She's asked him out on a date a couple of times even though she knows he is with him. One time she even called him asking to go out to lunch and when he said no she started crying about how he's supposed to be with and how it's not fair and stuff. I really wish my boyfriend would just cut her off because I worry sometimes that he'll listen to her and go back to her. He gets really mad though when I say I want him to just cut her off and I don't get why he hasn't yet. He's worried that I'm going to start controlling who he's friends with all around. (That's something she did when or they were dating). He also says he doesn't want to be petty by blocking her and stuff. Any advice on getting him to cut her off? Or any advice on just dealing with that fact they will still talk. 
P.S. He says he never actually replies to the messages that she sends.