2000/11/30 by Marc A. Rieffel · 1 citation
Mathematics · Physics and Astronomy · #math.OA #hep-th #math.MG #quant-ph #msc:46L87 #msc:58B30 #msc:60B10
published as Mem. Amer. Math. Soc. 168 (2004) no. 796, 1-65 · 81 pages. Several minor improvements and several references added. To appear Memoirs Amer. Math. Soc
arxiv created 2003/02/20 · arxiv updated 2009/11/30
By a quantum metric space we mean a C^*-algebra (or more generally an order-unit space) equipped with a generalization of the Lipschitz seminorm on functions which is defined by an ordinary metric. We develop for compact quantum metric spaces a version of Gromov-Hausdorff distance. We show that the basic theorems of the classical theory have natural quantum analogues. Our main example involves the quantum tori, A_þ. We show, for consistently defined ``metrics'', that if a sequence \þn\ of parameters converges to a parameter þ, then the sequence \Aþn\ of quantum tori converges in quantum Gromov-Hausdorff distance to A_þ.