I'm running Ubuntu 12.04. In the Vuze downloads properties, it says
Owner: root. I don't know how to change that. This has only happened in Ubuntu, not Windows.
You were probably running Vuze as root, so all files downloaded by it are now owned by root. Make sure Vuze is closed, open a terminal and go to the folder where the files in question are located. Then run the following command:
sudo chown -cR $USER:$(id -gn $USER) .
(Please note that the point at the end is part of the command). Now start Vuze and everything should work just fine.