An Example where distGH > distq

Let (A,LA) be a quantum metric space. Then clearly S(A) with the metric ρLA is a compact balanced convex metric space, i.e. the metric ρLA is convex and balanced. Another important property of S(A) is that the R−valued affine continuous functions Af(S(A)) separate the points of S(A). Conversly, let (X, ρ) be a compact balanced convex metric space on which Af(X) separate the points of X. Then Af(X) is an order-unit space. By some standard argument involving separating disjoint compact convex subsets, we see X = S(Af(X)) and the metric topology coincides with the w−topology. By Theorem 9.7 in [1], ρ comes from a norm M on (Af(X)) = {f ∈ (Af(X)) : f(1Af(X)) = 0} = R(X − X). Let Af(X)L = (Lipρ) ∩ Af(S(A)). By Lemma 9.9 in [1], Af(X)L is dense in Af(X). So S(Af(X)L) = X, and the w −topology on X as the states space of Af(X)L is the same as that from Af(X) hence the same as the metric topology. By Theorem 9.8 in [1], ρ on X is induced by the Lip-norm Lρ on Af(X)L. Therefore (Af(X)L, Lρ) is a quantum metric space and (X, ρ) is its states space. So the states spaces of quantum metric spaces are exactly compact balanced convex metric spaces whose affine coutinuous functions separate ponits.