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A Jordan GNS Construction for the Holonomy-Flux *-algebra

Michael Rios
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Abstract

The holonomy-flux *-algebra was recently proposed as an algebra of basic kinematical observables for loop quantum gravity. We show the conventional GNS construction breaks down when the the holonomy-flux *-algebra is allowed to be a Jordan algebra of observables. To remedy this, we give a Jordan GNS construction for the holonomy-flux *-algebra that is based on trace. This is accomplished by assuming the holonomy-flux *-algebra is an algebra of observables that is also a Banach algebra, hence a JB algebra. We show the Jordan GNS construction produces a state that is invariant under all inner derivations of the holonomy-flux *-algebra. Implications for the corresponding Jordan-Schrodinger equation are also discussed.

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Michael Rios. 2005-05-09. A Jordan GNS Construction for the Holonomy-Flux *-algebra. https://arxiv.org/abs/gr-qc/0505038

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