Connes spectral distance - Imported Knowlpedia Content

Definition

Let $(\mathcal A,H,D)$ be a unital spectral triple, and let $\varphi,\psi$ be states on the $C^*$ -completion of $\mathcal A$ . Their Connes spectral distance is

d_D(\varphi,\psi)= \sup\left\{ |\varphi(a)-\psi(a)|: a=a^*,\quad a\in\mathcal A, L_D(a)\leq1 \right\},

where $L_D$ is the spectral Lipschitz seminorm. The value may be $+\infty$ . It is an extended pseudometric in general and becomes an extended metric when the kernel of $L_D$ on self-adjoint elements consists only of scalars and the algebra separates states.

Why it is a distance

Symmetry and the triangle inequality follow from the absolute value and linearity of states. If $L_D(a)=0$ , every scalar multiple of $a$ remains admissible. Hence two states differing on such an element are at infinite distance. If all zero-seminorm self-adjoint elements are scalar, equality $d_D(\varphi,\psi)=0$ forces the states to agree on the dense algebra and therefore on its completion.

Finiteness does not imply that $d_D$ induces the weak-star topology on the state space. In the compact quantum metric setting, one additionally asks for the Lipschitz unit ball modulo scalars to be suitably totally bounded Rieffel, §§1–2.

Recovery of geodesic distance

For the canonical spin spectral triple of a connected closed Riemannian spin manifold $M$ , evaluation at $x\in M$ defines a pure state $\delta_x$ . Because $\lVert[\not D,f]\rVert$ is the supremum norm of $df$ , the dual formula for geodesic distance gives

d_{\not D}(\delta_x,\delta_y)=d_{\mathrm{geo}}(x,y).

Thus the operator $D$ and its commutators recover the original metric on points without referring to coordinates Connes, Chapter VI, §1.

On the full state space of $C(M)$ , the same supremum gives the Kantorovich–Rubinstein distance between probability measures. This extension contains more metric information than the restriction to pure states.

Conventions and scope

Using all $a\in\mathcal A$ instead of only self-adjoint $a$ gives the same value for states when the seminorm is star-invariant, but the self-adjoint formula makes reality explicit. Some authors impose $\lVert a\rVert\leq1$ as an extra cutoff in noncompact settings; that produces a bounded variant, not the distance defined in the core.

References

A. Connes, Noncommutative Geometry, Academic Press, 1994. Author-hosted text. Relevant: Chapter VI, §1, especially the distance formula for the canonical commutative triple.
M. A. Rieffel, “Metrics on State Spaces,” Documenta Mathematica 4 (1999), 559–600. DOI record. Relevant: §§1–2 on state-space metrics defined by Lipschitz seminorms.