Ask yourself how you'd feel if she walked into a room and found you reading her texts. Would you be ashamed of yourself? Would you feel like you shouldn't be doing it? Do you think she'd be angry? If the answer to any of those questions is yes, then don't do it.
I don't think I'd mind too much if my boyfriend read my texts because I have nothing to hide. There might occasionally be messages from my ex, but I'm sure he gets messages from his ex from time to time. But I've never been tempted to read his texts, and I don't think I would because his phone is private and I don't have an automatic right to see it. I wouldn't want him to walk into a room and find me reading his texts, so that tells me I shouldn't do it. I'm pretty certain he's not cheating on me, and although I can't prove it, at some point you have to either trust someone, or if you can't, break up with them.