The present Memoir consists of two closely related papers. The first paper introduces a notion of Gromov-Hausdorff distance for quantum metric spaces and establishes its basic properties, which turn out to be natural analogs of the classical ones. By a quantum metric space the author means a \(C^*\)-algebra equipped with a generalization of the usual Lipschitz seminorm on functions which one associates to an ordinary metric. The main example presented here is that of the quantum tori equipped with an appropriate metric structure. It is shown that the quantum Gromov-Hausdorff distance between two quantum tori is a continuous function of the traditional parameters specifying the tori.
The main result of the second paper is convergence (with respect to the quantum Gromov-Hausdorff distance) of the appropriately metrized full matrix algebras to the \(C^*\)-algebra of continuous functions on an integral coadjoint orbit of a compact semisimple Lie group. This result provides a precise meaning for assertions scattered in the theoretical physics literature saying that the sequence of full matrix algebras of increasing dimensions converges to the 2-sphere.
Both papers are nicely written and may serve as an introduction to the theory of Gromov-Hausdorff distance for \(C^*\)-algebras, developed by the author and other specialists.