  The operator algebraic framework plays an important role in mathematical
physics. Many different operator algebras exist for example for a theory of
quantum mechanics. In Loop Quantum Gravity only two algebras have been
introduced until now. In the project about 'Algebras of Quantum Variables (AQV)
for LQG' the known holonomy-flux *-algebra and the Weyl C*-algebra will be
modified and a set of new algebras will be proposed and studied. The idea of
the construction of these algebras is to establish a finite set of operators,
which generates (in the sense of Woronowicz, Schm\"udgen and Inoue) the
different O*- or C*-algebras of quantum gravity and to use inductive limits of
these algebras. In the Loop Quantum Gravity approach usually the basic
classical variables are connections and fluxes. Studying the three constraints
appearing in the canonical quantisation of classical general relativity in the
ADM-formalism some other variables like curvature appear. Consequently the main
difficulty of a quantisation of gravity is to find a suitable replacement of
the set of elementary classical variables and constraints. The algebra of
quantum gravity is supposed to be generated by a set of the operators
associated to holonomies, fluxes and in some cases even the curvature. There
are two reasonable choices for this algebra: The set of constraints of Quantum
Gravity are contained in or at least the constraints are affilliated with this
algebra. Secondly, the algebra of quantum variables is said to be physical if
it contains complete observables. In the project of 'Algebras of Quantum
Variables for LQG' different algebras will be studied with respect to the
property of being a physical algebra. Furthermore the existence of KMS-states
on these algebras will be argued.
