Why don't we see them in MMOs?
There are two primary aspects to an arena: competition and spectators. MMOs fail to design for spectators. These games are full of battles and competitions -- PvE, PvP, RvR -- but audiences who can watch these battles while remaining outside the conflict are purely incidental when they do exist.
Predictably, my closest experience to playing spectator in an MMO was in early Star Wars: Galaxies. While sitting in a cantina so that an entertainer could heal my fatigue, a gunfight suddenly sprang up between a Rebel and an Imperial soldier. It was exactly like the cantina scene in the first Star Wars film: for the duration of the fight, the music and dancing stopped and all eyes were upon the warriors. When the battle was over, the music and dancing resumed; the audience returned to their socializing.
Here's where I answer my own question. Arenas have obvious potential for fun: imagine players rooting for and betting on their favorite fighters and favorite teams, or players capturing mobs to sell to the arena, etc. But we don't see arenas in MMOs because dynamics and skill are downplayed in the standard MMO combat template.
I'm fine with games being designed for gamers who are not like me. I have plenty of friends and family who can only enjoy a game if the control complexity and dependence on skill are less than I generally prefer. MMOs reach many people like this, among others. But it's sad that such avoidance of depth and dynamics in MMO combat so completely dominates the genre that features such as arenas are impossible.
But maybe I'm wrong. Do you think arenas are possible with current MMO standards?