Abstract
Under some reasonable assumptions on a spectral triple \( (\mathcal{A}^{\infty }, \mathcal{H}, D) \) (which we call admissibility), we prove the existence of a universal object in the category of compact quantum groups admitting coaction on the closure of \( \mathcal{A}^{\infty } \) which commutes with the (noncommutative) Laplacian. This universal object is called the quantum isometry group w.r.t. the Laplacian. Moreover, we discuss analogous formulations of the quantum group of orientation (and volume or a given real structure) preserving isometries. Sufficient conditions under which the action of the quantum isometry group keeps the \( C^* \) algebra invariant and is a \( C^* \) action are given. We also mention some sufficient conditions for the existence of the quantum group of orientation preserving isometries without fixing a choice of the ‘volume-form’.
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