With a ATI Radeon HD 5670 run starcraft 2 on high?

Depends on the resolution. But, performance is not going to be great. Using anti-aliasing will result in poor performance. See following review:

http://www.xbitlabs.com/articles/graphics/display/starc...

Why do you not spend a little to get a starting range Gaming graphics card HD 5750 or even HD 5770?

Short awnser, yes on low resolution, no on high resolution.
But for the price of an iMac you can get a way better gameing rig....

1680 x 1050 high settings:

What will the primary purpose of the computer be? If you need the Mac for specific reasons, then sure. If gaming is your objective, then you can do better.

Thank you all for your help. I have decided to buy a newer refurbished iMac. The reason i wanted that HD 5670 model was because it was on Ebay for $800. Instead i bought a new 27" Quad Core i5 3.1GHz processor with 16 gigs of ram with an AMD Radeon HD 6970M with the refurbished deal from apple. ordinarily this would be a $2000 machine but i saved $400 and got it for $1600. overall the reason i am buying a Macintosh computer is because i am starting college for a film degree. I wanted to use what the industry uses while still being able to play videogames. I am very happy with my purchase and while it was hard to get use to mac i think i got it all down now. Best of all i can play any game i want while running windows in Bootcamp. i would definitely recommend people to see if apple brand computers are in your price range because while you must pay a premium, it does a lot of things(like gaming) a lot better than people think.

Also on another note while iMacs are not really upgradable in anyway, with the new thunderbolt port and the possibly of a 3rd party external GPU unit i could upgrade the graphics card.

If that's the case, then its not bad.