  The Weyl algebra A of continuous functions and exponentiated fluxes,
introduced by Ashtekar, Lewandowski and others, in quantum geometry is studied.
It is shown that, in the piecewise analytic category, every regular
representation of A having a cyclic and diffeomorphism invariant vector, is
already unitarily equivalent to the fundamental representation. Additional
assumptions concern the dimension of the underlying analytic manifold (at least
three), the finite wide triangulizability of surfaces in it to be used for the
fluxes and the naturality of the action of diffeomorphisms -- but neither any
domain properties of the represented Weyl operators nor the requirement that
the diffeomorphisms act by pull-backs. For this, the general behaviour of
C*-algebras generated by continuous functions and pull-backs of homeomorphisms,
as well as the properties of stratified analytic diffeomorphisms are studied.
Additionally, the paper includes also a short and direct proof of the
irreducibility of A.
