Much of the work in loop quantum gravity and quantum geometry rests on a
mathematically rigorous integration theory on spaces of distributional
connections. Most notably, a diffeomorphism invariant representation of the
algebra of basic observables of the theory, the Ashtekar-Lewandowski
representation, has been constructed. This representation is singled out by its
mathematical elegance, and up to now, no other diffeomorphism invariant
representation has been constructed. This raises the question whether it is
unique in a precise sense.
In the present article we take steps towards answering this question. Our
main result is that upon imposing relatively mild additional assumptions, the
AL-representation is indeed unique.
As an important tool which is also interesting in its own right, we introduce
a C*-algebra which is very similar to the Weyl algebra used in the canonical
quantization of free quantum field theories.