Quantum symmetries of classical spaces

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We give a general scheme for constructing faithful actions of genuine (noncommutative as C* algebra) compact quantum groups on classical topological spaces. Using this, we show that: (i) a compact connected classical space can have a faithful action by a genuine compact quantum group, and (ii) there exists a spectral triple on a classical connected compact space for which the quantum group of orientation and volume preserving isometries (in the sense of qorient) is a genuine quantum group.