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, is the quantum Gromov–Hausdorff limit of any sequence C∗ Hn, n n∈N for the natural quantum metric structures and when the lifts of n to G converge pointwise to . 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. © 2005 Elsevier Inc. All rights reserved.