Noncommutative $C^k$ functions and Fr\'{e}chet derivatives of operator functions

Fix a unital $C^*$-algebra $\mathscr{A}$, and write $\mathscr{A}_{sa}$ for the set of self-adjoint elements of $\mathscr{A}$. Also, if $f:\mathbb{R}\to\mathbb{C}$ is a continuous function, then write $f_\mathscr{A}:\mathscr{A}_{sa}\to\mathscr{A}$ for the operator function $a\mapsto f(a)$ defined via functional calculus. In this paper, we introduce and study a space $NC^k(\mathbb{R})$ of $C^k$ functions $f:\mathbb{R}\to\mathbb{C}$ such that, no matter the choice of $\mathscr{A}$, the operator function $f_\mathscr{A}:\mathscr{A}_{sa}\to\mathscr{A}$ is $k$-times continuously Fr\'echet differentiable. In other words, if $f\in NC^k(\mathbb{R})$, then $f$ "lifts" to a $C^k$ map $f_\mathscr{A}:\mathscr{A}_{sa}\to\mathscr{A}$, for any (possibly noncommutative) unital $C^*$-algebra $\mathscr{A}$. For this reason, we call $NC^k(\mathbb{R})$ the space of noncommutative $C^k$ functions. Our proof that $f_\mathscr{A}\in C^k(\mathscr{A}_{sa};\mathscr{A})$, which requires only knowledge of the Fr\'echet derivatives of polynomials and operator norm estimates for "multiple operator integrals" (MOIs), is more elementary than the standard approach; nevertheless, $NC^k(\mathbb{R})$ contains all functions for which comparable results are known. Specifically, we prove that $NC^k(\mathbb{R})$ contains the homogeneous Besov space $\dot{B}_1^{k,\infty}(\mathbb{R})$ and the H\"older space $C_{loc}^{k,\varepsilon}(\mathbb{R})$. We highlight, however, that the results in this paper are the first of their type to be proven for arbitrary unital $C^*$-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 $W_k(\mathbb{R})_{loc}\subsetneq NC^k(\mathbb{R})\subsetneq C^k(\mathbb{R})$, where $W_k(\mathbb{R})_{loc}$ is the "localized" $k^{th}$ Wiener space.