Smooth domain of the Dirac derivation - Imported Knowlpedia Content

Definition

Let $\delta(T)=[|D|,T]$ be the Dirac derivation on $B(H)$ . Its smooth domain is

\operatorname{Dom}\delta^\infty =\bigcap_{k\geq1}\operatorname{Dom}\delta^k.

Thus a bounded operator $T$ lies in $\operatorname{Dom}\delta^\infty$ precisely when every iterated commutator

\delta^k(T)=[|D|,[|D|,\ldots,[|D|,T]\ldots]]

initially defined on the appropriate common domain, extends to a bounded operator on $H$ . The notation $\mathrm{OP}^0$ is often used for this same smooth operator algebra in the pseudodifferential calculus associated with a spectral triple. Membership is an all-orders regularity condition, not merely boundedness of the first commutator.

Fréchet algebra structure

The seminorms

q_k(T)=\|\delta^k(T)\|,\qquad k=0,1,2,\ldots,

where $\delta^0(T)=T$ , define a complete locally convex topology on the smooth domain. The Leibniz rule expresses $\delta^k(ST)$ as a finite sum of products of iterated derivatives of $S$ and $T$ , so $\operatorname{Dom}\delta^\infty$ is an algebra. It is also closed under adjoints because $\delta(T^*)=-\delta(T)^*$ .

Role in spectral regularity

A spectral triple is regular exactly when its represented algebra and all basic commutators $[D,a]$ lie in this smooth domain. This hypothesis permits repeated commutator expansions and serves as the order-zero coefficient algebra for the local-index pseudodifferential calculus Connes–Moscovici, §II.

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 on the smooth domain of $\delta$ and abstract pseudodifferential operators.
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: §§3–4 on regularity and differential operator algebras.