Compact Quantum Metric Spaces

We give a brief survey of many of the high-lights of our present understanding of the young subject of quantum metric spaces, and of quantum Gromov-Hausdorff distance between them. We include examples. My interest in developing the theory of compact quantum metric spaces was stimulated by certain statements in the high-energy physics and string-theory literature, concerning non-commutative spaces that converged to other (possibly noncommutative) spaces. These statements appeared to me to deserve a more precise formulation. (See the references in the introductions of [33] [34].) Here I will just give a brief survey of some of the main developments in this very young subject. I will also indicate some of the main classes of examples which have been explored so far. I include a few related arguments which are not quite in place in the existing literature. 1. The definition of compact quantum metric spaces The concept of a quantum metric space has its origins in Connes’ paper [8] of 1989 in which he first proposes using Dirac operators as the vehicle for metric data in non-commutative geometry. He was motivated by his observation that for a compact spin Riemannian manifold one can recover its smooth structure, its Riemannian metric, and much else, directly from its standard Dirac operator. This led him to the concept of a spectral triple, (A,H, D), consisting of a ∗-algebra A represented by bounded operators on a Hilbert space H, and of a (usually unbounded) self-adjoint operator, D, on H such that the commutator [D, a] is a bounded operator for each a ∈ A. Connes also requires that D have compact resolvant. (Spectral triples are very closely related to “unbounded K-cycles” or “unbounded Fredholm modules”, the difference being that for spectral triples the representation of A must be faithful.) Connes pointed out that from a spectral triple one obtains a metric, ρD, on the state space, S(A), of A by means of the formula ρD(μ, ν) = sup{|μ(a)− ν(a)| : ‖[D, a]‖ ≤ 1}, 2000 Mathematics Subject Classification. Primary 46L87, 14E20; Secondary 53C23, 58B34.