Update HT6030 (http://support.apple.com/kb/HT6030) was supposed to fix GMail issues with the Mavericks Mail.app. But now I see the same issue as in earlier versions before setting the "All Mail" label to be invisible in GMail's IMAP settings:
After setting it back to visible (which is necessary to have the archive option working in the Mail.app), Notification Center shows all incoming messages twice, although the Mail.app is set to display only messages from the "Inbox" in the Dock and Notification center.
Thanks and beste
MacBook Air (13-inch Mid 2012), OS X Mavericks (10.9.2)