Finiteness and absolute-continuity axiom for a spectral triple

Definition

Let $(\mathcal A,H,D)$ be a $p$ -dimensional regular spectral triple and set $H^\infty=\bigcap_{k\geq1}\operatorname{Dom}D^k$ . It satisfies the finiteness and absolute-continuity axiom when $H^\infty$ is a finitely generated projective left $\mathcal A$ -module and carries an $\mathcal A$ -valued Hermitian pairing $(\cdot\mid\cdot)_{\mathcal A}$ for which

\langle\xi,a\eta\rangle_H = \int_D a\,(\xi\mid\eta)_{\mathcal A}

for all $a\in\mathcal A$ and $\xi,\eta\in H^\infty$ . Here $\int_D$ is the chosen noncommutative integral, normally defined from the critical power $|D|^{-p}$ . Thus algebraic finite-projectivity and analytic volume compatibility are both required.

Geometric meaning

For the canonical spin spectral triple of a closed $p$ -dimensional Riemannian spin manifold, $H^\infty$ is the module of smooth spinor sections. The Serre–Swan theorem makes this module finitely generated projective over $C^\infty(M)$ . Its pointwise Hermitian product is $C^\infty(M)$ -valued, and integration against Riemannian volume recovers the $L^2$ -inner product. The axiom abstracts precisely these two facts.

Finiteness prevents the smooth domain from behaving like an arbitrary infinite-rank module. Absolute continuity ties its Hilbert-space completion to the same volume functional that the spectrum of $D$ determines.

Normalization and sidedness

The displayed identity uses a left module and the convention that the Hilbert-space inner product is linear in its second variable. With a right-module convention, the order of $a$ and the module pairing changes. Sources may also multiply the noncommutative integral by a dimension-dependent constant. These are convention changes, not extra geometric axioms.

The formula requires more than the statement that $H^\infty$ is a finitely generated projective module. One must specify a positive Hermitian module structure and verify that its integrated pairing equals the given Hilbert-space product.

Role in reconstruction

Finiteness and absolute continuity are independent of regularity, Hochschild orientability, and the first-order condition. In reconstruction proofs, finite projectivity produces a smooth vector bundle once the algebra has been identified with $C^\infty(X)$ , while absolute continuity identifies $H$ with its $L^2$ -space of sections Rennie–Várilly, §§3.1 and 7.

References

A. Connes, Noncommutative Geometry, Academic Press, 1994. Author-hosted text. Relevant: Chapter VI, §1, finiteness and absolute continuity among the spectral axioms.
A. Rennie and J. C. Várilly, “Reconstruction of Manifolds in Noncommutative Geometry,” 2007. Stable preprint. Relevant: §3.1 for the axiom system and §7 for reconstruction of the smooth bundle and Hilbert space.