Higher derivatives of operator functions in ideals of von Neumann algebras

Abstract

Let $\mathscr{M}$ be a von Neumann algebra and $a$ be a self-adjoint operator affiliated with $\mathscr{M}$. We define the notion of an "integral symmetrically normed ideal" of $\mathscr{M}$ and introduce a space $OC^{[k]}(\mathbb{R}) \subseteq C^k(\mathbb{R})$ of functions $\mathbb{R} \to \mathbb{C}$ such that the following result holds: for any integral symmetrically normed ideal $\mathscr{I}$ of $\mathscr{M}$ and any $f \in OC^{[k]}(\mathbb{R})$, the operator function $\mathscr{I}_{\mathrm{sa}} \ni b \mapsto f(a+b)-f(a) \in \mathscr{I}$ is $k$-times continuously Fréchet differentiable, and the formula for its derivatives may be written in terms of multiple operator integrals. Moreover, we prove that if $f \in \dot{B}_1^{1,\infty}(\mathbb{R}) \cap \dot{B}_1^{k,\infty}(\mathbb{R})$ and $f'$ is bounded, then $f \in OC^{[k]}(\mathbb{R})$. Finally, we prove that all of the following ideals are integral symmetrically normed: $\mathscr{M}$ itself, separable symmetrically normed ideals, Schatten $p$-ideals, the ideal of compact operators, and -- when $\mathscr{M}$ is semifinite -- ideals induced by fully symmetric spaces of measurable operators.