Semifinite local index formula - Imported Knowlpedia Content

Statement

Let $(\mathcal A,\mathcal N,D,\tau)$ be a regular, finitely tau-summable semifinite spectral triple with isolated dimension spectrum and the analytic-continuation hypotheses of Carey–Phillips–Rennie–Sukochev. The semifinite local index formula represents its Chern character by a finite residue cocycle built from $\tau$ , elements $a_j$ , commutators $[D,a_j]$ , and iterated commutators with $D^2$ . Pairing this cocycle with $K_1(\mathcal A)$ in odd parity or $K_0(\mathcal A)$ in even parity equals the Breuer-index pairing. In odd parity it also computes semifinite spectral flow from $D$ to $uDu^*$ .

Residue structure

The formula replaces ordinary traces in the Connes–Moscovici local index formula by the chosen faithful normal semifinite trace $\tau$ . Its cochains are finite linear combinations of residues of zeta functions having the schematic form

z\longmapsto \tau\!\left( a_0[D,a_1]^{(k_1)}\cdots[D,a_m]^{(k_m)} (1+D^2)^{-z} \right),

where $T^{(k)}$ denotes the $k$ -fold commutator with $D^2$ . Universal coefficients and residue points depend on parity, degree, and normalization; the schematic expression is not a substitute for the full formulas.

Part I derives the odd formula through spectral flow Carey–Phillips–Rennie–Sukochev, Theorem 4.1 and §8. Part II proves the even formula and a generalized McKean–Singer identity Carey–Phillips–Rennie–Sukochev, §§4–6.

Meaning and consequences

The bounded-transform pairing is defined globally through a Breuer–Fredholm compression. The residue cocycle computes the same class using finitely many commutators and spectral asymptotics. This makes index pairings accessible to heat-kernel, pseudodifferential, and zeta-function calculations even when $\mathcal N$ has nontrivial center and the index is real-valued rather than integral.

When $\mathcal N=B(H)$ and $\tau$ is the ordinary trace, the formula specializes to the usual local index formula under the corresponding hypotheses. The extension is substantive: tau-compactness and tau-summability can hold even when the relevant operators are not compact or summable in the ordinary Hilbert-space sense.

Conventions and scope

The theorem identifies cyclic-cohomology classes or their pairings; differently normalized residue representatives may differ by coboundaries. Sign conventions for spectral flow and the odd index pairing must be coordinated.

References

A. L. Carey, J. Phillips, A. Rennie, and F. A. Sukochev, “The Local Index Formula in Semifinite von Neumann Algebras I: Spectral Flow,” Advances in Mathematics 202 (2006), 451–516. Stable preprint. Relevant: Theorem 4.1 and §§7–8, the odd residue cocycle and spectral-flow pairing.
A. L. Carey, J. Phillips, A. Rennie, and F. A. Sukochev, “The Local Index Formula in Semifinite von Neumann Algebras II: The Even Case,” Advances in Mathematics 202 (2006), 517–554. DOI record. Relevant: §§4–6, the even residue cocycle and generalized McKean–Singer formula.