Do I have a right to not trust my boyfriend?

So my boyfriend has cheated on my in the beginning of our relationship. He was in another relationship and had sex with her. He also was asking other girls to be his girlfriend and made out with another girl while dating me. He had phone sex. I caught him and now he is changing his behavior. He went to therapy and gave me access to his accounts. The thing that worries me is now he isn’t putting any effort in the relationship. He won’t call me iniate any conversations first. Won’t fly out to see me because he’s saying he’s broke and doesn’t have the money. He doesn’t iniate anything sexual and he always has an excuse. He said he’s withdrawing because of how much I mistrust him. He did this to his ex girlfriend too. He said he did to her because he didn’t care and that he’s doing it to me because I’m mistreating him and making him feel like I’m going to abandon him. Should I trust by now or am I giving good reasons not to trust him?
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Do I have a right to not trust my boyfriend?
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