Higher derivatives of operator functions in ideals of von Neumann algebras

Let M be a von Neumann algebra and self-adjoint operator affiliated with M. We define the notion of an “integral symmetrically normed ideal” introduce space OC[k](R)⊆Ck(R) functions R→C such that following holds: for any integral ideal I f∈OC[k](R), function Isa∋b↦f(a+b)−f(a)∈I is k-times continuously Fréchet differentiable, formula its derivatives may written in terms multiple integrals. Moreover, we prove if f∈B˙11,∞(R)∩B˙1k,∞(R) f′ bounded, then f∈OC[k](R). Finally, all ideals are normed: itself, separable ideals, Schatten p-ideals, compact operators, – when semifinite induced by fully symmetric spaces measurable operators.