It is possible to administrate spaces if a user is a space administrator. This allows not only Confluence administrators to clean spaces, but also space administrators.
To access Space Administrator page, just click the 'Space tool' menu in the bottom left corner of the page and then click the section 'Cleaner for Confluence'
The Space Admin page looks very similar to the Admin page; the only difference is that the space selected is the current space where the user is Space Admin and Is only possible to select the current space.
