Fix a unital $C^{\ast }$-algebra $\mathcal{A}$, and write ${\mathcal{A}}_{\mathrm{sa}}$ for the set of self-adjoint elements of $\mathcal{A}$. Also, if $f:\mathbb{R}\to ℂ$ is a continuous function, then write $f_{\mathcal{A}}:{\mathcal{A}}_{\mathrm{sa}}\to \mathcal{A}$ for the operator function $a\mapsto f\left(a\right)$ defined via functional calculus. In this paper, we introduce and study a space $N{C}^k\left(\mathbb{R}\right)$ of $C^k$ functions $f:\mathbb{R}\to ℂ$ such that, no matter the choice of $\mathcal{A}$, the operator function $f_{\mathcal{A}}:{\mathcal{A}}_{\mathrm{sa}}\to \mathcal{A}$ is $k$-times continuously Fréchet differentiable. In other words, if $f\in N{C}^k\left(\mathbb{R}\right)$, then $f$ “lifts” to a $C^k$ map $f_{\mathcal{A}}:{\mathcal{A}}_{\mathrm{sa}}\to \mathcal{A}$, for any (possibly noncommutative) unital $C^{\ast }$-algebra $\mathcal{A}$. For this reason, we call $N{C}^k\left(\mathbb{R}\right)$ the space of noncommutative $C^k$ functions. Our proof that $f_{\mathcal{A}}\in C^k({\mathcal{A}}_{\mathrm{sa}};\mathcal{A})$, 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, $N{C}^k\left(\mathbb{R}\right)$ contains all functions for which comparable results are known. Specifically, we prove that $N{C}^k\left(\mathbb{R}\right)$ contains the homogeneous Besov space ${\dot{B}}_1^{k,\infty }\left(\mathbb{R}\right)$ and the Hölder space $C_{\mathrm{loc}}^{k,\varepsilon }\left(\mathbb{R}\right)$. 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 $W_k{\left(\mathbb{R}\right)}_{\mathrm{loc}}⊊N{C}^k\left(\mathbb{R}\right)⊊C^k\left(\mathbb{R}\right)$, where $W_k{\left(\mathbb{R}\right)}_{\mathrm{loc}}$ is the “localized” $k$th Wiener space.

修理单位$C^{＊}$-代数$\mathcal{A}$， 和写${\mathcal{A}}_{\mathrm{萨}}$对于自伴随元素的集合$\mathcal{A}$. 另外，如果$F:\mathbb{R}\to ℂ$是连续函数，则写$F_{\mathcal{A}}:{\mathcal{A}}_{\mathrm{萨}}\to \mathcal{A}$对于运算符函数 $A\mapsto F\left(A\right)$通过泛函定义。在本文中，我们介绍并研究了一个空间$否C^k\left(\mathbb{R}\right)$的$C^k$功能$F:\mathbb{R}\to ℂ$这样，无论选择$\mathcal{A}$, 运算符函数$F_{\mathcal{A}}:{\mathcal{A}}_{\mathrm{萨}}\to \mathcal{A}$是$k$- 次连续 Fréchet 可微。换句话说，如果$F\in 否C^k\left(\mathbb{R}\right)$， 然后$F$“提升”到$C^k$地图$F_{\mathcal{A}}:{\mathcal{A}}_{\mathrm{萨}}\to \mathcal{A}$，对于任何（可能是非交换的）单元$C^{＊}$-代数$\mathcal{A}$. 为此，我们称$否C^k\left(\mathbb{R}\right)$非交换空间 $C^k$ 功能。我们的证明$F_{\mathcal{A}}\in C^k({\mathcal{A}}_{\mathrm{萨}};\mathcal{A})$，它只需要了解多项式的 Fréchet 导数和“多算子积分”（MOI）的算子范数估计，比标准方法更基本；尽管如此，$否C^k\left(\mathbb{R}\right)$包含已知可比较结果的所有函数。具体来说，我们证明$否C^k\left(\mathbb{R}\right)$包含齐次 Besov 空间${\dot{乙}}_{\mathrm{1个}}^{k,\infty }\left(\mathbb{R}\right)$和霍尔德空间$C_{\mathrm{位置}}^{k,\epsilon }\left(\mathbb{R}\right)$. 然而，我们强调，本文中的结果是第一个被证明适用于任意单元的类型$C^{＊}$-algebras，并且这种一般设置的扩展利用了作者最近通过 MOI 的定义解决的某些“可分离性问题”。最后，我们通过展示具体例子来证明$W_k{\left(\mathbb{R}\right)}_{\mathrm{位置}}⊊否C^k\left(\mathbb{R}\right)⊊C^k\left(\mathbb{R}\right)$， 在哪里$W_k{\left(\mathbb{R}\right)}_{\mathrm{位置}}$是“本地化”$k$维纳空间。