Approximation of quantum tori by finite quantum tori for the quantum Gromov-Hausdorff distance

We establish that, given a compact Abelian group G endowed with a continuous length function l and a sequence (Hn)n∈N of closed subgroups of G converging to G for the Hausdorff distance induced by l, then C*G^,σ is the quantum Gromov-Hausdorff limit of any sequence C*Hn^,σnn∈N for the natural quantum metric structures and when the lifts of σn to G^ converge pointwise to σ. This allows us in particular to approximate the quantum tori by finite-dimensional C*-algebras for the quantum Gromov-Hausdorff distance. Moreover, we also establish that if the length function l is allowed to vary, we can collapse quantum metric spaces to various quotient quantum metric spaces.