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In the project AQV the issue of quantum constraints, KMS-states and algebrasof quantum configuration and momentum variables in Loop Quantum Gravity hasbeen argued. There a physical algebra has been required to contain completeobservables and the quantum constraints, or at least the quantum constraintsare affilliated with this algebra. In this context a first conjecture for aphysical algebra is presented in this article. A new *-algebra for LGQ, whichis called the localised holonomy-flux cross-product *-algebra, is studied. Asuggestion for a physical *-algebra, which contains the localised holonomy-fluxcross-product *-algebra, a modified quantum Hamilton constraint, a localisedquantum diffeomorphism constraint and even a modified quantum Masterconstraint, is given.

AQV V. The localised holonomy-flux cross-product *-algebra