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Motivated by the work of Goswami on quantum isometry groups of non‐commutative manifolds, we define the quantum symmetry group of a unital  C *‐algebra A equipped with an orthogonal filtration as the universal object in the category of compact quantum groups acting on A in a filtration‐preserving fashion. The existence of such a universal object is proved and several examples discussed. In particular, we study the universal quantum group acting on the dual of the free group and preserving both the word length and the block length.

Quantum symmetry groups of C *‐algebras equipped with orthogonal filtrations