AQV IV. A new formulation of the holonomy-flux *-algebra

In this article the holonomy-flux *-algebra, which has been introduced by Lewandowski, Okolow, Sahlmann and Thiemann, is modificated. The new *-algebra is called the holonomy-flux cross-product *-algebra. This algebra is an abstract cross-product *-algebra. It is given by the universal algebra of the algebra of continuous and differentiable functions on the configuration space of generalised connections and the universal enveloping flux algebra associated to a surface set, and some canonical commutator relations. There is a uniqueness result for a certain path- and graph-diffeomorphism invariant state of the holonomy-flux cross-product *-algebra. This new *-algebra is not the only *-algebra, which is generated by the algebra of certain continuous and differentiable functions on the configuration space of generalised connections and the universal enveloping flux algebra associated to a surface set. The theory of abstract cross-product algebras allows to define different new *-algebras. Some of these algebras are presented in this article.