Spectral Lipschitz seminorm - Imported Knowlpedia Content

Definition

Let $(\mathcal A,H,D)$ be a spectral triple, with $\mathcal A$ represented on $H$ . Its spectral Lipschitz seminorm is

L_D(a)=\lVert[D,a]\rVert,\qquad a\in\mathcal A,

where $[D,a]$ denotes its bounded extension. On a larger ambient $C^*$ -algebra one may set $L_D(a)=+\infty$ when no bounded extension exists. The triangle inequality and Leibniz estimate

L_D(ab)\leq L_D(a)\lVert b\rVert+\lVert a\rVert L_D(b)

make $L_D$ an extended Leibniz seminorm. It can vanish on nonscalar elements commuting with $D$ , so it need not be a norm modulo the scalars.

Metric role

The unit ball $\{a=a^*:L_D(a)\leq1\}$ is the set over which the Connes spectral distance takes its supremum. Only differences of states are evaluated, so adding a scalar to $a$ does not change that distance. If the kernel of $L_D$ on the self-adjoint part consists exactly of scalars, the induced quotient seminorm separates states; additional compactness conditions are needed for its metric to induce the weak-star topology Rieffel, §§1–2.

The seminorm isolates the first-order metric information of a spectral triple. Summability, grading, real structure, and regularity may affect other parts of the geometry but are not ingredients in this definition.

Canonical example

For the canonical spin spectral triple of a closed Riemannian spin manifold $M$ , Clifford multiplication gives

[\not D,f]=c(df),\qquad L_{\not D}(f)=\sup_{x\in M}|df_x|.

Thus $L_{\not D}$ is the classical Lipschitz seminorm on smooth functions, and its completion recovers ordinary Lipschitz functions. This identity is the analytic input behind the recovery of Riemannian geodesic distance Connes, Chapter VI, §1.

A bounded operator commuting with $D$ has seminorm zero, even when it is not scalar. Such a commutant element is therefore a decisive obstruction to obtaining a genuine metric on all states.

Conventions and scope

Some authors define $L_D$ only on the self-adjoint part of a dense order-unit space; others define it on the complex algebra and restrict only when constructing a metric. “Lipschitz ball” may mean the unit ball itself, its image modulo scalars, or a norm-bounded slice. These sets have different compactness properties.

For a nonunital or locally compact spectral triple, the unit ball is usually too large to be compact without a choice of base state, localization, or additional properness condition. The formula for $L_D$ remains meaningful, but compact quantum metric-space conclusions do not follow automatically.

References

A. Connes, Noncommutative Geometry, Academic Press, 1994. Author-hosted text. Relevant: Chapter VI, §1 on the distance formula and Dirac commutator norm.
M. A. Rieffel, “Metrics on State Spaces,” Documenta Mathematica 4 (1999), 559–600. DOI record. Relevant: §§1–2 on Lipschitz seminorms, quotient seminorms, and metrics on state spaces.