how to stop an email sender from continually downloading the messages from archive mailbox. I have accidentally activated and it won't stop downloading

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After force quitting the application, restart your MacBook Air. If after quitting the app and restarting the Mac the issue remains, reply back and let me know.
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This is an example for gmail web server account .See this article Archive or mute Gmail messages - Computer - Gmail Help

If you want to clean up your inbox without deleting your emails, you can archive or mute them. Your emails are moved to a label called "All Mail."

• When you archive a message: The message will come back to your inbox when someone replies to it.
• When you mute a message: Any replies stay out of your inbox. You can search for the conversation if you want to find it again.

And later on you can create a filter for a particular sender Create rules to filter your emails - Gmail Help

As what ever mails come in inbox on the web server are mirrored in apple mail application , you can do only changes in web mail account .

Also - Messages you archived aren't deleted, and you can still find them any time.