[doppelganger]

I think you're probably against another ace here most of the time, usually with a better kicker. Any broadway card will probably give you a split, but could complete a straight draw for a non-A hand, so I would probably bet to protect my weak ace here. I'd probably bet 1/2 pot to take away drawing odds but still be able to get away from the hand if someone raises.

Edit: having read the responses, it doesn't make sense to bet since any draws are looking for 4 outs and are unlikely anyway.