Unbounded Representatives of the Chern–Connes Character

Definitions and mathematical statement

Unbounded-character construction question. Let $(A,H,D)$ be a $p$-summable odd spectral triple and write $F=D|D|^{-1}$. Can the Chern--Connes character of the associated Fredholm module be constructed using only $D$ and commutators $[D,a]$, $a\in A$? The requested meaning of using only must be specified; because $F$ is obtained from $D$ by functional calculus, simply reusing a bounded-phase formula under that description would not explain the intended unbounded representative.

In a conventional unital setup, $A$ is an involutive algebra represented by bounded operators on $H$, $D$ is densely defined self-adjoint with compact resolvent, and each $[D,a]$ on its initial domain extends to a bounded operator. Odd means there is no imposed even grading. One strict summability convention is $(1+D^2)^{-1/2}\in\mathcal L^p(H)$, where $\mathcal L^p$ is the Schatten ideal; weak-ideal endpoint conventions differ and are not identified silently. In a nonunital version the compactness and summability conditions are localized by elements of $A$. The source does not state which convention is intended.

The phase formula requires a kernel convention. If $D$ is invertible, the displayed inverse is defined in the usual sense. Otherwise one must specify a pseudoinverse on $(\ker D)^\perp$ and a choice on the finite-dimensional kernel, or a permitted finite-rank modification. Without this, $|D|^{-1}$ is not an everywhere-defined inverse. The associated odd Fredholm module has bounded phase commuting with $A$ modulo compact operators in the required sense. Its Chern--Connes character conventionally lies in odd periodic cyclic cohomology, represented by appropriate trace cocycles in a degree compatible with summability.

The task is to construct a well-defined unbounded-data representative of that same character, and state its cyclic or cocycle identities and equality of classes, under the source's precise hypotheses. It must specify whether functional calculus, resolvents, heat operators, regularized traces, or residues are permitted, and whether the target is periodic or another cyclic theory. No smooth regularity, dimension-spectrum, or trace-continuation hypothesis needed by a particular local formula is inserted. The source supplies none of these extra requirements, so no universal formula is invented.

The supplied triage checked 2026-08-21 labels the extracted question open and reports no located complete resolution, with the report as its sole citation. This dated assessment does not identify the exact class of permitted formulas and requires independent verification.

Short English statement

Seek a cyclic-character representative expressed through the unbounded operator and its commutators, with the same class as the bounded Fredholm module. The allowable functional calculus, summability convention, cyclic theory, and kernel treatment are missing, so merely writing $F$ as a function of $D$ is not taken as the requested construction.

Sources

• Local UnsolvedMath registry, permanent ID 30000620, code catalogue entry; Unbounded Representatives of the Chern--Connes Character, Oberwolfach Reports Open Problems.
• Mini-workshop: Dirac Operators in Differential and Noncommutative Geometry (2007), Oberwolfach Reports, DOI 10.4171/owr/2006/53, pp. 3131--3188. https://doi.org/10.4171/owr/2006/53. Proposed year and year tag: 2006; no proposer named.
• The same full report is the sole supplied literature source; no separate precise unbounded-character theorem is supplied.
• Literature triage checked 2026-08-21, status open; record updated 2026-09-16. The corrected_verified field reports source checking, not independently verified here.