p-summable spectral triple - Imported Knowlpedia Content

Definition

Let $0<p<\infty$ . A spectral triple $(\mathcal A,H,D)$ is $p$ -summable if

(1+D^2)^{-1/2}\in\mathcal L^p(H),

where $\mathcal L^p(H)$ is the Schatten class. Equivalently,

\operatorname{Tr}\!\left((1+D^2)^{-p/2}\right)<\infty.

This is a quantitative compact-resolvent condition: the singular values of the regularized inverse of $D$ are $p$ -summable, with multiplicity. The regularization by $1+D^2$ treats $\ker D$ harmlessly. For an invertible $D$ , it may be replaced by $|D|^{-1}$ without changing membership in the relevant ideal.

Equivalent spectral tests

When $D$ has compact resolvent and eigenvalues $\lambda_n$ , repeated with multiplicity, $p$ -summability is equivalent to

\sum_n(1+\lambda_n^2)^{-p/2}<\infty.

It is also equivalent to finiteness of the spectral zeta function $\operatorname{Tr}(|D|^{-p})$ after omitting the finite-dimensional kernel, provided $D$ is invertible away from zero. These equivalences follow directly from functional calculus and the definition of Schatten ideals Gracia-Bondía–Várilly–Figueroa, §10.1.

If the triple is $p$ -summable, then it is $q$ -summable for every $q>p$ . The converse can fail at the endpoint, so the value $p$ must not be inferred merely from summability at all larger exponents.

Examples and consequences

The canonical spin spectral triple on a closed $n$ -dimensional manifold is $p$ -summable for every $p>n$ , by Weyl eigenvalue asymptotics. At the critical exponent $p=n$ , its inverse Dirac operator generally belongs only to a weak Schatten ideal, while its $n$ th power supports a Dixmier trace. Thus saying that the triple has dimension $n$ is not the same as saying it is strictly $n$ -summable.

Finite summability implies theta summability because exponential decay dominates every negative power. It also makes sufficiently long products of Dirac or bounded-transform commutators trace class, enabling finite-degree cyclic Chern characters.

Conventions and scope

For $0<p<1$ , the same trace formula defines membership in a quasi-Banach Schatten ideal rather than a Banach ideal. This extension is included in the core convention and in the linked Schatten-class convention. Semifinite spectral triples replace the canonical operator trace by a specified faithful normal semifinite trace.

References

A. Connes, Noncommutative Geometry, Academic Press, 1994. Author-hosted text. Relevant: Chapter IV, §§1–2 on summable Fredholm modules and trace ideals.
J. M. Gracia-Bondía, J. C. Várilly, and H. Figueroa, Elements of Noncommutative Geometry, Birkhäuser, 2001. DOI record. Relevant: §10.1 on summability of spectral triples.