Higher derivatives of operator functions in ideals of von Neumann algebras

Let M be a von Neumann algebra and a be a self-adjoint operator affiliated with M. We define the notion of an "integral symmetrically normed ideal" of M and introduce a space OC[k](R) subset of C-k(R) of functions R -> C such that the following holds: for any integral symmetrically normed ideal I of M and any f is an element of OC[k](R), the operator function I-sa (sic) b (sic) f(a + b) - f(a) is an element of I is k-times continuously Frechet differentiable, and the formula for its derivatives may be written in terms of multiple operator integrals. Moreover, we prove that if f is an element of B-1(1,infinity) (R) boolean AND B-1(k,infinity) (R) and f ' is bounded, then f is an element of OC[k](R). Finally, we prove that all of the following ideals are integral symmetrically normed: M itself, separable symmetrically normed ideals, Schatten p-ideals, the ideal of compact operators, and - when M is semifinite - ideals induced by fully symmetric spaces of measurable operators. (c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).