I think it depends on the child.
My little brother is 12 and my parents bought him Black Ops. Obviously it's a fun game and, although very violent, I believe my brother certainly knows it's not real and he doesn't take any of it to heart (who knows, perhaps he plays 'war' in the playground, but who didn't at that age, COD or no COD?).
If this 11 year old is intelligent then I think he would understand and it should be fine. Having said this, my little brother has two old brothers, one 16 and me, 20 - therefore he's grown up with us doing these things, perhaps making him more numb to the effects.
It's hard to say. I know when I was 11 the worse game I was playing was Tekken on the PS1, but that promotes fighting!?
In conclusion, I have to say that it all depends on the 11 year old to whether they should play it. Obviously, they shouldn't - but this isn't a perfect world and they do. It's whether or not you want them to miss out on the part of their childhood where they go into school and discuss with their friends how much they 'owned' last night and what time they're going to play that night.
I also think it promotes teamwork and builds friendships (although you must make them go out and play football with his friends sometimes, keep them fit and all that!)