Noncommutative $L_p$-differentiability and trace formulae

Abstract

Let $\cM$ be a semifinite von Neumann algebra equipped with a normal faithful semifinite trace $τ$, and let $L_p(\cM)$ denote the associated noncommutative $L_p$-space for $1<p<\infty$. Let $n\in\N$ and let $a, b$ be self-adjoint operators affiliated with $\cM$, with $b\in L_p(\cM)\cap L_{np}(\cM)$. For a function $f\in C^n(\R)$ whose derivatives $f^{(k)}$ are bounded for $1\le k\le n$, we prove that the map $φ:t\in\R\mapsto f(a+tb)-f(a)$ is $n$-times differentiable in the $\|\cdot\|_{L_p}$-norm. This strengthens the corresponding result of de Pagter and Sukochev for $p\neq 2$ and extends it to higher-order derivatives. In addition, if $b\in \cM$, then $φ^{(n)}$ is continuous on $\R$. Consequently, we extend the Potapov--Skripka--Sukochev higher-order trace formula from bounded $L_n(\cM)$-perturbations to not necessarily bounded perturbations in $L_n(\cM)\cap L_{n(n+1)}(\cM)$. Moreover, we show that this trace formula holds for a broader class of admissible functions than the classes previously considered in the literature.

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• Definition 2.2
• Theorem 2.3
• Remark 2.4
• Lemma 2.5
• Corollary 2.6