  Let g be a (say, sufficiently differentiable) function on the reals. One
knows how to apply g to Hermitian elements A of a C* algebra. Yet the question
of differentiability of the mapping A to g(A) is not trivial, since in general
"A and dA do not commute". However, since this mapping depends linearly on g,
one can, via Fourier transform, reduce the case of general g to the case of the
exponential function. For the latter one has an explicit formula for the n-th
derivative (more complicated than in the scalar case - still "A and dA do not
commute"). In this way one bounds the norm of the n-th derivative of (A to
g(A)) on the r-ball by a Sobolev norm involving the (n+1)-th derivative of g on
[-r,r].

Operator Algebras;Functional Analysis