Noncommutative $L_p$-differentiability and trace formulae

Let $\mathcal{M}$ be a semifinite von Neumann algebra equipped with a normal faithful semifinite trace $\tau$, and let $L_p(\mathcal{M})$ denote the associated noncommutative $L_p$-space for $1<p<\mathrm{\infty }$. Let $n\in \mathbb{N}$ and let $a,b$ be $\tau$-measurable self-adjoint operators such that $b\in L_p(\mathcal{M})\cap L_{np}(\mathcal{M})$. For a function $f\in C^n(\mathbb{R})$ whose derivatives $f^{(k)}$ are bounded for $1\le k\le n$, we prove that the map $\phi :t\in \mathbb{R}\mapsto f(a+tb)-f(a)$ is $n$-times differentiable in the $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{L_p}$-norm. This strengthens the corresponding result of de Pagter and Sukochev for $p\mathrm{\ne }2$ and extends it to higher-order derivatives. In addition, if $f^{(n)}\in C_0(\mathbb{R})$ or $b\in \mathcal{M}$, then ${\phi }^{(n)}$ is continuous on $\mathbb{R}$. Consequently, we extend the Potapov–Skripka–Sukochev higher-order trace formula from bounded $L_n$-perturbations to not necessarily bounded perturbations in $L_n(\mathcal{M})\cap L_{n^2}(\mathcal{M})$. Moreover, we show that this trace formula holds for a broader class of admissible functions than the classes previously considered in the literature.