Tau-summable semifinite spectral triple - Imported Knowlpedia Content

Definition

Let $(\mathcal A,\mathcal N,D,\tau)$ be a semifinite spectral triple, where $\tau$ is a faithful normal semifinite trace on $\mathcal N$ , and let $p>0$ . The triple is tau- $p$ -summable if

(1+D^2)^{-1/2}\in L^p(\mathcal N,\tau),

equivalently,

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

Here $D$ is self-adjoint and affiliated with $\mathcal N$ , its algebra commutators are bounded, and its resolvent is tau-compact. Thus summability is measured by generalized singular values and $\tau$ , not by ordinary eigenvalue multiplicities. For a nonunital algebra, the local convention usually requires $a(1+D^2)^{-1/2}\in L^p(\mathcal N,\tau)$ for every $a\in\mathcal A$ instead.

Equivalent decay formulation

Writing $\mu_t$ for the generalized singular-value function, tau- $p$ -summability is equivalent to

\int_0^\infty \mu_t\!\left((1+D^2)^{-1/2}\right)^p\,dt<\infty.

Weak tau- $p$ -summability replaces $L^p$ by the corresponding weak ideal and permits the borderline decay $O(t^{-1/p})$ . The strong and weak conditions are not interchangeable; the latter is the natural endpoint in many dimension- $p$ examples Carey, Phillips, Rennie, and Sukochev, §2.

Examples and conventions

For $\mathcal N=B(H)$ with the usual trace, $L^p(\mathcal N,\tau)$ is the Schatten class, so the definition is ordinary $p$ -summability. Finite trace alone does not make every affiliated resolvent $p$ -summable: the displayed trace must still be finite.

Some authors call the triple $p$ -summable when $(1+D^2)^{-s/2}$ is trace-class for every $s>p$ , reserving “ $p^+$ -summable” or “weakly $p$ -summable” for the critical endpoint. Any assertion of metric dimension should state which convention is in force.

References

A. L. Carey, J. Phillips, A. Rennie, and F. A. Sukochev, “The Hochschild Class of the Chern Character for Semifinite Spectral Triples,” Journal of Functional Analysis 213 (2004), 111–153. DOI record. Relevant: §2 on semifinite spectral triples and $(p,\infty)$ -summability.
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. DOI record. Relevant: §2 on trace ideals, generalized singular values, and summability.