Quantum Isometry Groups of Discrete Quantum Spaces

Abstract

We show that the definitions of quantum symmetry groups for finite metric spaces and graphs given by Banica and Bichon can be viewed as quantum isometry groups in our sense. Next, we prove that the quantum group of orientation preserving isometries of a spectral triple on some approximately finite dimensional \( C^* \) algebras arise as the inductive limit of the quantum group of orientation preserving isometries of suitable spectral triples on the constituent finite dimensional algebras.