Just an addition to my question.
I changed the access rights one my back-up volume such that now everyone has read and write access.
To my surprise the time machine browser now opens.
However, preparing a back-up now takes very long. I cancelled the preparation twice.
I will add a new comment, when back-ups will work again.
Just the last addition to my question.
The modifications of the access rights took very, very long. In the end I cancelled the process.
Nevertheless, time machine now works correctly. As far as I can see it.
If annybody runs into similar problems. The solution is I think just to modify the access rights on the back-up volume to read and write access for everyone.
Probably OS X Lion added an undocumented feature to the time machine.