Finitely summable spectral triple - Imported Knowlpedia Content

Definition

A spectral triple $(\mathcal A,H,D)$ is finitely summable if it is $p$ -summable for at least one finite exponent $p>0$ ; explicitly, there exists $p<\infty$ such that

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

Equivalently, the eigenvalue counting function of $|D|$ has sufficiently slow polynomial growth for some finite power of its regularized inverse to be trace class. The exponent is not part of the data of finite summability. Once one exponent works, every larger exponent works as well.

What the condition controls

Finite summability is stronger than compact resolvent: an arbitrary compact operator can have singular values that fail to belong to every finite Schatten class. It is also stronger than theta summability, because polynomial summability implies convergence of every positive-time heat trace, while the converse need not hold.

The condition supplies finite-degree trace cocycles. If the commutators of the associated bounded Fredholm module lie in $\mathcal L^p$ , then products of sufficiently many commutators are trace class and define its cyclic Chern character Connes, Chapter IV, §1.

Examples and non-examples

Every canonical Dirac spectral triple on a closed finite-dimensional manifold is finitely summable, by Weyl asymptotics. A diagonal operator on $\ell^2(\mathbb N)$ with eigenvalues growing like $\log(n+1)$ has compact resolvent but is not finitely summable, since

\sum_n\bigl(1+\log^2(n+1)\bigr)^{-p/2}

diverges for every finite $p$ . This example isolates the failed property: compactness holds, but no polynomial trace-ideal bound does.

Finite-dimensional spectral triples are automatically finitely summable for every $p$ , but finite summability does not mean that $H$ , $\mathcal A$ , or the spectrum is finite.

Conventions and scope

Some sources use “finite summability” for the weaker assertion that the regularized inverse lies in a weak Schatten ideal of finite order. The core uses strict trace-class powers, in agreement with the linked $p$ -summability convention. When only weak endpoint behavior is known, the ideal and exponent should be stated.

Finite summability says nothing by itself about regularity, dimension spectrum, or meromorphic continuation of zeta functions. Those are separate hypotheses in the local index formula.

References

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