Regular spectral triple - Imported Knowlpedia Content

Definition

A spectral triple $(\mathcal A,H,D)$ is regular if

\mathcal A+[D,\mathcal A]\subseteq \operatorname{Dom}\delta^\infty, \qquad \delta(T)=[|D|,T].

Here $\mathcal A+[D,\mathcal A]$ denotes the linear span of represented operators $a$ and bounded commutators $[D,b]$ , and $\operatorname{Dom}\delta^\infty$ is the smooth domain of the Dirac derivation. Equivalently, for every $a\in\mathcal A$ , all iterated commutators $\delta^k(a)$ and $\delta^k([D,a])$ extend to bounded operators for every integer $k\geq1$ . Regularity is an additional differentiability axiom; it does not follow from the bounded first-commutator axiom of a spectral triple.

Analytic consequences

Regularity supports a filtered algebra of abstract differential and pseudodifferential operators. In that calculus, repeated commutators with $|D|$ behave like derivatives, while operators in $\operatorname{Dom}\delta^\infty$ behave like order-zero coefficients. With separate summability and meromorphic-continuation hypotheses, this machinery leads to residues of zeta functions and the local index formula Connes–Moscovici, §II.

Standard example

For the canonical spin spectral triple of a closed Riemannian spin manifold, smooth functions and their commutators with the Dirac operator remain bounded under all iterated commutators with $|D|$ . This follows from the classical pseudodifferential calculus. By contrast, merely completing the smooth function algebra in the operator norm generally introduces nonsmooth functions and destroys this all-orders condition.

Conventions and scope

References

A. Connes and H. Moscovici, “The Local Index Formula in Noncommutative Geometry,” Geometric and Functional Analysis 5 (1995), 174–243. DOI record. Relevant: §II, regularity and the pseudodifferential calculus.
N. Higson, “The Local Index Formula in Noncommutative Geometry,” in Contemporary Developments in Algebraic K-Theory, ICTP Lecture Notes 15, 2004. Author-hosted manuscript. Relevant: smoothness hypotheses and the differential-operator approach.