Compact quantum metric space - Imported Knowlpedia Content

Definition

Let $A$ be a real order-unit space with order unit $e$ and state space $S(A)$ . A compact quantum metric space is a pair $(A,L)$ in which $L$ is a seminorm on a dense order-unit subspace of $A$ , $L(a)=0$ exactly for $a\in\mathbb Re$ , and

\rho_L(\varphi,\psi)=\sup\{|\varphi(a)-\psi(a)|:L(a)\leq1\}

induces the weak-star topology on $S(A)$ . The seminorm $L$ is then called a Lip-norm. For a unital $C^*$ -algebra $B$ , one takes $A=B_{\mathrm{sa}}$ ; the elements of $S(A)$ are the restrictions of states on $B$ .

Total-boundedness criterion

The weak-star metrizability condition is equivalent to total boundedness of the image of the $L$ -unit ball in the quotient $A/\mathbb Re$ . Equivalently, after choosing a state $\varphi_0$ , the slice

\{a\in A:L(a)\leq1,\ \varphi_0(a)=0\}

is totally bounded in the order-unit norm. This criterion is often more practical than checking the topology directly Rieffel, §2.

Spectral triples

A unital spectral triple can supply $L(a)=\lVert[D,a]\rVert$ . Its associated Connes spectral distance has the same dual formula as $\rho_L$ . It defines a compact quantum metric space only when the zero-seminorm elements are precisely the scalars and the resulting metric induces the weak-star topology. Compact resolvent of $D$ alone does not imply either condition Rieffel, §§1–2.

Classical example and a near miss

For a compact metric space $X$ , take $A=C(X,\mathbb R)$ and let $L$ be the ordinary Lipschitz seminorm. The resulting metric on states is the Kantorovich metric on probability measures and induces their weak-star topology. Thus compact metric spaces embed contravariantly into this framework.

If $L$ vanishes on a nonconstant function, it is not a Lip-norm: states that distinguish that function are infinitely far apart after rescaling it. The failed axiom is the scalar-kernel condition.

Conventions and scope

Rieffel's original definition is formulated for order-unit spaces, not only $C^*$ -algebras. Later frameworks often impose lower semicontinuity, a Leibniz inequality, or matrix-level conditions; these are additional structures, not part of the definition in the core. The term “Lip-norm” names $L$ , whereas “compact quantum metric space” names the pair $(A,L)$ .

References

Marc A. Rieffel, “Metrics on State Spaces,” Documenta Mathematica 4 (1999), 559–600. EMS DOI record. Relevant: §§1–2 on metrics induced by Lipschitz seminorms.
Marc A. Rieffel, Gromov–Hausdorff Distance for Quantum Metric Spaces, Memoirs of the American Mathematical Society 168, no. 796 (2004), 1–65. AMS DOI record. Relevant: §2 on compact quantum metric spaces and Lip-norm criteria.