Derivation δ(T) = [|D|, T] - Imported Knowlpedia Content

Definition

Let $(\mathcal A,H,D)$ be a spectral triple. The delta derivation is the generally unbounded derivation on $B(H)$ defined by

\delta(T)=[|D|,T].

Its domain consists of bounded operators $T$ that preserve $\operatorname{Dom}|D|$ and for which this operator commutator, initially defined on $\operatorname{Dom}|D|$ , extends to a bounded operator on $H$ . The bounded extension is by definition $\delta(T)$ . With the operator norm on $B(H)$ , $\delta$ is a closed derivation and satisfies $\delta(ST)=\delta(S)T+S\delta(T)$ whenever $S$ and $T$ lie in its domain.

Iterated domains and regularity

Define

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

A spectral triple is regular when every represented $a\in\mathcal A$ and every bounded commutator $[D,a]$ belongs to $\operatorname{Dom}\delta^\infty$ . Thus regularity means boundedness of every iterated commutator with $|D|$ , not merely boundedness of $[D,a]$ . This condition supplies the operator analogue of smooth coefficients needed by the pseudodifferential calculus in the local index formula Connes–Moscovici, §II.

Graph norms and smooth operator algebra

The graph-norm seminorms

q_k(T)=\|\delta^k(T)\|,\qquad k\geq0,

give $\operatorname{Dom}\delta^\infty$ a natural Fréchet algebra topology. Closedness of $\delta$ makes the successive graph norms complete. This topology records more information than the ambient operator norm and is the appropriate home for asymptotic expansions involving repeated commutators.

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 regular spectral triples and the derivation by $|D|$ .
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: smooth domains of the commutator derivation and the local-index pseudodifferential calculus.