Automorphism covariant representations of the holonomy-flux lowast-algebra

We continue the analysis of representations of cylindrical functions and fluxes which are commonly used as elementary variables of loop quantum gravity. We consider an arbitrary principal bundle of a compact connected structure group and, following Sahlmann's ideas (Sahlmann 2002 Preprint gr-qc/0207111), define a holonomy-flux ∗-algebra whose elements correspond to the elementary variables. There exists a natural action of automorphisms of the bundle on the algebra; this action generalizes the action of analytic diffeomorphisms and gauge transformations on the algebra considered in earlier works. We define the automorphism covariance of a ∗-representation of the algebra on a Hilbert space and prove that the only Hilbert space admitting such a representation is a direct sum of the spaces L 2 , given by a unique measure on the space of generalized connections. This result is a generalization of our previous work (Okołów and Lewandowski 2003 Class. Quantum Grav. 20 3543–67 ( Preprint gr-qc/0302059)) where we assumed that the principal bundle is trivial and its base manifold is .