Approximation of the spectral action functional in the case of $\tau$-compact resolvents

TL;DR

This paper develops estimates for the spectral action functional's Taylor remainders in the setting of semifinite von Neumann algebras, extending previous results to broader classes of functions and resolvent conditions.

Contribution

It extends existing spectral action approximation results to include broader function classes and resolvent conditions in noncommutative geometry.

Findings

Derived estimates for Taylor remainders of spectral action functional

Extended results to noncompactly supported functions with decay

Improved bounds when resolvent belongs to noncommutative L^n-space

Abstract

We establish estimates and representations for the remainders of Taylor approximations of the spectral action functional $V\mapsto\tau(f(H_0+V))$ on bounded self-adjoint perturbations, where $H_0$ is a self-adjoint operator with $\tau$-compact resolvent in a semifinite von Neumann algebra and $f$ belongs to a broad set of compactly supported functions including $n$-times differentiable functions with bounded $n$-th derivative. Our results significantly extend analogous results in \cite{SkAnJOT}, where $f$ was assumed to be compactly supported and $(n+1)$-times continuously differentiable. If, in addition, the resolvent of $H_0$ belongs to the noncommutative $L^n$-space, stronger estimates are derived and extended to noncompactly supported functions with suitable decay at infinity.