How do I delete the completed file download mark (the green check next to an item) from the search results?

You don't. That mark means that you already have the file so it serves to remind you not to download it again. Sometimes this stays as an open arrowed folder and sometimes reverts to a number of stars when completed ... not sure why but ... don't worry about it.

I have on occasions seen checkmarks against files I have not downloaded, when I have downloaded one the same with a different bit rate, and this could present a problem. As far as I know re-starting limewire will remove any, somehow checked in error, though many arrows will remain, probably until/unless you rename the file.

If this doesn't exactly answer your question, please let me know.

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