p-summable Fredholm module - Imported Knowlpedia Content

Definition

Let $p>0$ , and let $(H,\pi,F)$ be a normalized Fredholm module over a complex involutive algebra $\mathcal A$ , so $F=F^*$ , $F^2=I$ , and $[F,\pi(a)]$ is compact for every $a\in\mathcal A$ . The module is $p$ -summable when

[F,\pi(a)]\in\mathcal L^p(H) \qquad\text{for every }a\in\mathcal A,

where $\mathcal L^p(H)$ is the Schatten ideal. Thus summability strengthens compactness by prescribing an $\ell^p$ decay rate for the commutator singular values.

Parity and the Chern character

Summability is compatible with either even or odd parity. If $n$ has the module's parity and $n>p-1$ , Hölder's inequality for Schatten ideals makes the standard conditional-trace formula well defined:

\phi_n(a_0,\ldots,a_n) =\lambda_n\operatorname{Tr}_0\!\left( \Gamma\pi(a_0)[F,\pi(a_1)]\cdots[F,\pi(a_n)]\right).

Here $\Gamma$ is the grading operator in the even case and $I$ in the odd case, $\lambda_n$ is the chosen standard normalization constant, and

\operatorname{Tr}_0(T) =\tfrac12\operatorname{Tr}\!\bigl(F(FT+TF)\bigr).

This cocycle represents the Chern character of the Fredholm module. By contrast, ordinary Schatten Hölder makes the displayed product itself trace class only when $n\geq p$ . The resulting class lies in periodic cyclic cohomology and pairs with $K$ -theory to recover the Fredholm index.

Algebra and topology matter

Summability is normally required on a specified dense smooth subalgebra $\mathcal A$ of a represented $C^*$ -algebra, not automatically on every element of its norm completion. The property can therefore change when the chosen smooth algebra changes. For a Banach or locally convex algebra one also requires the representation and commutator map to have the appropriate continuity.

If the condition holds at exponent $p$ , it holds at every exponent $q>p$ , because $\mathcal L^p\subset\mathcal L^q$ . The least viable exponent, when it exists, records the dimension-like decay of the cycle.

Conventions and scope

For an unnormalized Fredholm module, authors may impose Schatten conditions on the self-adjointness and involutivity defects as well as on commutators; the core avoids this ambiguity by using a normalized representative. Some sources say “ $p$ -summable” when commutators lie only in the weak ideal $\mathcal L^{p,\infty}$ . Here $p$ -summable means strict Schatten membership. When $0<p<1$ , $\mathcal L^p$ is a quasi-Banach ideal, but the membership condition remains meaningful.

Do not confuse a $p$ -summable Fredholm module with a $p$ -summable spectral triple: bounded transformation relates the two notions, but the summability exponent of the resulting commutators requires a separate estimate.

References

A. Connes, Noncommutative Geometry, Academic Press, 1994. Author-maintained text. Relevant: Chapter IV, §1 on finitely summable Fredholm modules and their characters.
A. Connes, “Non-Commutative Differential Geometry,” Publications Mathématiques de l'IHÉS 62 (1985), 41–144. DOI record. Relevant: §§II.1–II.3 on summable Fredholm modules and cyclic cocycles.
N. Higson and J. Roe, Analytic K-Homology, Oxford University Press, 2000. Publisher record. Relevant: bounded Fredholm modules and analytic $K$ -homology.