Noncommutative Ck functions and Fréchet derivatives of operator functions

Fix a unital $C^{\ast }$-algebra $A$, and write $A_{\mathrm{sa}}$ for the set of self-adjoint elements of $A$. Also, if $f:R\to ℂ$ is a continuous function, then write $f_A:A_{\mathrm{sa}}\to A$ for the operator function $a\mapsto f\left(a\right)$ defined via functional calculus. In this paper, we introduce and study a space $N{C}^k\left(R\right)$ of $C^k$ functions $f:R\to ℂ$ such that, no matter the choice of $A$, the operator function $f_A:A_{\mathrm{sa}}\to A$ is $k$-times continuously Fréchet differentiable. In other words, if $f\in N{C}^k\left(R\right)$, then $f$ “lifts” to a $C^k$ map $f_A:A_{\mathrm{sa}}\to A$, for any (possibly noncommutative) unital $C^{\ast }$-algebra $A$. For this reason, we call $N{C}^k\left(R\right)$ the space of noncommutative $C^k$ functions. Our proof that $f_A\in C^k(A_{\mathrm{sa}};A)$, which requires only knowledge of the Fréchet derivatives of polynomials and operator norm estimates for “multiple operator integrals” (MOIs), is more elementary than the standard approach; nevertheless, $N{C}^k\left(R\right)$ contains all functions for which comparable results are known. Specifically, we prove that $N{C}^k\left(R\right)$ contains the homogeneous Besov space ${\dot{B}}_1^{k,\infty }\left(R\right)$ and the Hölder space $C_{\mathrm{loc}}^{k,\varepsilon }\left(R\right)$. We highlight, however, that the results in this paper are the first of their type to be proven for arbitrary unital $C^{\ast }$-algebras, and that the extension to such a general setting makes use of the author’s recent resolution of certain “separability issues” with the definition of MOIs. Finally, we prove by exhibiting specific examples that $W_k{\left(R\right)}_{\mathrm{loc}}⊊N{C}^k\left(R\right)⊊C^k\left(R\right)$, where $W_k{\left(R\right)}_{\mathrm{loc}}$ is the “localized” $k$th Wiener space.