Show only the right hand side of the topbar to the logged in users

Hello, is that possible to Show only the right hand side of the topbar to the logged in users? I want to disallow the ability for the users to use the dashboard and the wordpress icon with the links, i only want them to edit their profile, see messages, login/logout. Is that possible to do it with a piece of code in functions.php? If it is possible i want the topbar to appear only if the user is logged in. Thanks