How To Mute
Open Gmail, select the offending email chain, and click "More" in the menu bar at the top of the screen. Here, you'll find options to mark the chain as unread, mark it as important, or mute. Hit the "Mute" button, and you're done.
From the Gmail app, use the three dots in the top right corner to pull up the same menu, with the same "Mute" button.
Google will automatically archive all existing and incoming messages in the thread—meaning they'll remain deep in the filing cabinet of the company’s brains, but won’t show up in your inbox. You won't get notifications for new messages either. Should you want to unmute the conversation in the future, just type a name or keyword into the search box, highlight the thread, navigate to “More,” and click “Unmute.”