We introduce the quantum Gromov-Hausdorff propinquity, a new distance between
quantum compact metric spaces, which extends the Gromov-Hausdorff distance to
noncommutative geometry and strengthens Rieffel's quantum Gromov-Hausdorff
distance and Rieffel's proximity by making *-isomorphism a necessary condition
for distance zero, while being well adapted to Leibniz seminorms. This work
offers a natural solution to the long-standing problem of finding a framework
for the development of a theory of Leibniz Lip-norms over C*-algebras.Comment: 49 Pages. This is the first half of 1302.4058v2, which has been
accepted in Trans. Amer. Math. Soc. The second half is now a different paper
entitled "Convergence of Fuzzy Tori and Quantum Tori for the quantum
Gromov-Hausdorff Propinquity: an explicit approach
