Semifinite spectral triple - Imported Knowlpedia Content

Definition

A semifinite spectral triple $(\mathcal A,\mathcal H,D;\mathcal M,\tau)$ consists of a unital involutive algebra $\mathcal A$ represented in a semifinite von Neumann algebra $\mathcal M\subseteq B(\mathcal H)$ , a faithful normal semifinite trace $\tau$ on $\mathcal M$ , and a densely defined self-adjoint operator $D$ affiliated with $\mathcal M$ . Each $a\in\mathcal A$ preserves $\operatorname{Dom}(D)$ , the commutator $[D,a]$ extends boundedly, and the resolvent scale belongs to the tau-compact ideal:

(1+D^2)^{-1/2}\in\mathcal K(\mathcal M,\tau).

Thus both measurability and compactness are internal to the traced algebra $(\mathcal M,\tau)$ .

Relation to ordinary spectral triples

Taking $\mathcal M=B(\mathcal H)$ with its canonical trace makes $\mathcal K(\mathcal M,\tau)$ the ordinary compact operators. The definition then becomes the usual compact spectral triple. For a general semifinite algebra, tau-compactness can be weaker than Hilbert-space compactness, and the trace provides real-valued dimensions of spectral projections. This replacement is the basis of the semifinite local index formula Carey–Phillips–Rennie–Sukochev, §2.

Fredholm and summability data

The bounded transform

F=D(1+D^2)^{-1/2}

is a Breuer–Fredholm operator modulo the tau-compact ideal, and its trace index replaces the integer-valued Fredholm index. Summability is also measured by $\tau$ : for example, $p$ -summability may require $(1+D^2)^{-p/2}$ to be tau-integrable Carey–Phillips–Rennie–Sukochev, §2. Regularity, dimension spectrum, grading, and real structure are additional axioms rather than consequences of semifiniteness.

Examples and variants

An ordinary spectral triple is the basic example via $\mathcal M=B(\mathcal H)$ . Geometric operators on a regular covering can instead be placed in the von Neumann algebra of equivariant operators and measured using its canonical semifinite trace, producing $L^2$ -type indices.

For nonunital $\mathcal A$ , global tau-compact resolvent is usually replaced by local compactness

a(1+D^2)^{-1/2}\in\mathcal K(\mathcal M,\tau) \qquad(a\in\mathcal A).

Authors also vary between $(1+D^2)^{-1/2}$ and resolvent formulations; for self-adjoint $D$ , functional calculus relates these conventions.

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. Preprint record. Relevant: §2 on semifinite spectral triples, tau-compactness, and Breuer–Fredholm operators.
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: the even semifinite framework and generalized McKean–Singer formula.