  In the project AQV the issue of quantum constraints, KMS-states and algebras
of quantum configuration and momentum variables in Loop Quantum Gravity has
been argued. There a physical algebra has been required to contain complete
observables and the quantum constraints, or at least the quantum constraints
are affilliated with this algebra. In this context a first conjecture for a
physical algebra is presented in this article. A new *-algebra for LGQ, which
is called the localised holonomy-flux cross-product *-algebra, is studied. A
suggestion for a physical *-algebra, which contains the localised holonomy-flux
cross-product *-algebra, a modified quantum Hamilton constraint, a localised
quantum diffeomorphism constraint and even a modified quantum Master
constraint, is given.
