On Quantum Duality of Group Amenability

In this paper, we investigate the co-amenability of compact quantum groups. Combining with some properties of regular C*-norms on algebraic compact quantum groups, we show that the quantum double of co-amenable compact quantum groups is unique. Based on this, this paper proves that co-amenability is preserved under formulation of the quantum double construction of compact quantum groups, which exhibits a type of nice symmetry between the co-amenability of quantum groups and the amenability of groups.

The classical-quantum duality of nature including gravity

The classical-quantum duality at the basis of quantum theory is here extended to the Planck scale domain. The classical/semiclassical gravity (G) domain is dual (in the precise sense of the classical-quantum duality) to the quantum (Q) elementary particle domain: [Formula: see text], [Formula: see text] being the Planck scale. This duality is universal. From the gravity (G) and quantum (Q) variables [Formula: see text], we define new quantum gravity (QG) variables [Formula: see text] which include all (classical, semiclassical and QG) domains passing through the Planck scale and the elementary particle domain. The QG variables are more complete than the usual ([Formula: see text], [Formula: see text]) ones which cover only one domain (Q or G). Two [Formula: see text] or [Formula: see text] values [Formula: see text] are needed for each value of [Formula: see text] (reflecting the two dual ways of reaching the Planck scale). We perform the complete analytic extension of the QG variables through analytic (holomorphic) mappings which preserve the light-cone structure. This allows us to reveal the classical-quantum duality of the Schwarzschild–Kruskal spacetime: exterior regions are classical or semiclassical while the interior is totally quantum: its boundaries being the Planck scale. Exterior and interior lose their difference near the horizon: four Planck scale hyperbolae border the horizons as a quantum dressing or width: “l’horizon habillé”. QG variables are naturally invariant under [Formula: see text]. Spacetime reflections, antipodal symmetry and PT or CPT symmetry are contained in the QG symmetry, which also shed insight on the global properties of the Kruskal manifold and its present renewed interest.