Hi Marko,
Many thx for your reply. I will do some tests, then I will be back with the result. I understand that the majority of WP use cases are Apache. Maybe I misunderstood the Total Cache doc statements and some web resources, and answers, that it supports IIS out of the box… It is not a problem if it does not, (again the mainstream is Apache), but I wasted hours because of trying to achieve this out of the box support…
Ron, many thanks for the answer.
Meanwhile it turned out that site url for that site in my MU installation was literally containing the .com ending (I mean: http://www.myname.family.com). I do not know I made a mistake when set it, or some automatism appended to it.
Anyway, now I had set it without the erratic .com ending (http://www.myname.family) and it works correctly.