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 (H-n)(n is an element of N) of closed subgroups of G converging to G for the Hausdorff distance induced by 1, then C* ((G) over cap, a) is the quantum Gromov-Hausdorff limit of any sequence C*((H) over cap (n), sigma(n))(n is an element of N) for the natural quantum metric structures and when the lifts of sigma(n) to (G) over cap converge pointwise to sigma. 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. (c) 2005 Elsevier Inc. All rights reserved.