As it stands messages aren't sent for combat in the arena. Since arena combat is so fast most of the time when I log in I just find my men sent back home with no idea who killed them or what the damage exchange was. It would be interesting to see this data (especially as non-arena combat is quite rare for me) in my battle messages.

That's strange. There's a 5 second lag for each attack/move on the arena. That's quire enough to analyze the attack log message.
If you mean you send the heroes to the arena and return in 20 minutes to see what happened to them, that's not the battle format the arena serves to. It's all about immediate action and quick responses.