Necessity is the mother of invention ;-)
After a lot of trial and error I took at look inside the /etc/apache2/sites-enabled folder, where ISPConfig saves the individual site configuration in files like this one:
Here I simply uncommented the line beginning with:
# php_admin_value open_basedir /var/www/clients/client2/web9/web:/var/www/clients/client2/web9/tmp...blablabla...
And eAccelerator started immediately the caching mechanism. Works like a charm and what a performance boost!
So there is maybe one question left now.
Can I live without php_admin_value open_basedir ...blablabla..?