I disagree. If the menu options clearly invite you to DTRT, then you
shouldn't have to worry about things like "the process is running". You
click on the menu, you start watching email, you get notifications, you
jump to your relevant email folder.
Mark
Yes and no.. I agree that you shouldn't have to worry about it, but there should still be some sort of visual indicator. Just as an example, the OS X dock leaves a little blue indicator below icons of processes that are running, but for which you have closed all windows. It does not affect the behaviour of the dock (clicking the icon will still either focus the most recent window or, if there are none open, start a new one), but the user can see the difference.
-Scott