In simple terms, it is the State of Registry
that decides what privileges (if any) a particular set of pilot papers confers on an aircraft of that Registry.
In this case it is the FAA, and the reference is FAR 61.3.
You don't say which country that PPL was issued
in, but under 61.3 it is legit to fly an N-reg in the airspace of that country alone
If the PPL was issued
in France then the pilot can fly an N-reg in French airspace only.
The fact that the PPL is JAR-FCL is not relevant.
It's really quite simple once you remember the first principle above.
Some notes here