Higher derivatives of operator functions in ideals of von Neumann algebras

Abstract

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)⊆Ck(R) of functions R→C such that the following holds: for any integral symmetrically normed ideal I of M and any f∈OC[k](R), the operator function Isa∋b↦f(a+b)−f(a)∈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∈B˙11,∞(R)∩B˙1k,∞(R) and f′ is bounded, then f∈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.