UC San Diego

A combination of the method of perturbation formulas and polynomial approximation is employed to compute the $k^{\text{th}}$ derivatives of (1) maps on symmetrically normed ideals of a unital Banach algebra induced by a holomorphic function, (2) maps on the self-adjoint elements of symmetrically normed ideals of a unital $C^*$-algebra induced by functions of a real variable that are ``slightly better than $C^k$,'' and (3) maps on the self-adjoint elements of integral symmetrically normed ideals of a von Neumann algebra induced by functions of a real variable that are ``slightly better than $C^k$ and Lipschitz.'' Along the way, the ``separation of variables'' approach to defining multiple operator integrals on non-separable Hilbert spaces is developed. As an application to free probability, a free It\^o formula of P. Biane and R. Speicher is extended, reinterpreted, and made more computationally flexible.