Is there a way to know if an item has been bought on the auction house?

Solution 1:

No, there is no way to know for sure if an item has been bought from the auction house.

Solution 2:

There is a way though to check if a specific item changed ownership but there are many limitations

from diabloprogress.com

they keep record of the items that have been seen on characters the only problem is that the item you see at AH and want to check must be an item that the player had equiped for some time so to be registered on him.. so if you see that this item changed ownership then you can assume that someone bought..

but as i told you this is very limited and not accurate 100% can't think any other way