Higher derivatives of operator functions in ideals of von Neumann algebras

Abstract

Let $\mathcal{M}$ be a von Neumann algebra and $a$ be a self-adjoint operator affiliated with $\mathcal{M}$. We define the notion of an "integral symmetrically normed ideal" of $\mathcal{M}$ and introduce a space $O{C}^{[k]}(\mathbb{R})\subseteq C^k(\mathbb{R})$ of functions $\mathbb{R}\to \mathbb{C}$ such that the following result holds: for any integral symmetrically normed ideal $\mathcal{I}$ of $\mathcal{M}$ and any $f\in O{C}^{[k]}(\mathbb{R})$, the operator function ${\mathcal{I}}_{\mathrm{sa}}\ni b\mapsto f(a+b)-f(a)\in \mathcal{I}$ 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 $f^{\prime }$ is bounded, then $f\in O{C}^{[k]}(\mathbb{R})$. Finally, we prove that all of the following ideals are integral symmetrically normed: $\mathcal{M}$ itself, separable symmetrically normed ideals, Schatten $p$-ideals, the ideal of compact operators, and -- when $\mathcal{M}$ is semifinite -- ideals induced by fully symmetric spaces of measurable operators.

43 pages. This version has been updated to match the published version, aside from the inclusion of a sketch of proof of Proposition 2.2.8 (omitted from the published version), the correction of some typos, and the adjustment of some references

References in corpus (1)

• Multiple operator integrals in non-separable von Neumann algebras

Cited by in corpus (2)

• Noncommutative $C^k$ functions and Fréchet derivatives of operator functions
• Multiple operator integrals in non-separable von Neumann algebras