论文标题
非交换性$ c^k $函数和操作员功能的fréchet衍生物
Noncommutative $C^k$ functions and Fréchet derivatives of operator functions
论文作者
论文摘要
修复一个Unital $ c^*$ - 代数$ \ Mathscr {a} $，并为$ \ Mathscr {a} $的sexwaxhixhodAdexhighatchoint Elements编写$ \ Mathscr {a} _ {sa} $。另外，如果$ f：\ mathbb {r} \ to \ mathbb {c} $是一个连续的函数，则写入$ f_ \ m mathscr {a}：\ mathscr {a} _ {sa} _ {sa} \ to \ to \ to \ to \ to \ nato \ a} $ a \ mapsto $ a \ mapsto f（a a）在本文中，我们介绍和研究$ c^k $ functions $ nc^k（\ mathbb {r}）$ f：\ mathbb {r} \ to \ mathbb {c} $，无论如何选择$ \ mathscr {a} $， $ f_ \ mathscr {a}：\ Mathscr {a} _ {sa} \ to \ mathscr {a} $是$ k $ - times nimes visherfréchet。换句话说，如果$ f \ in nc^k（\ mathbb {r}）$，则$ f $“ lifts”到a $ c^k $ map $ f_ \ mathscr {a}：\ mathscr {a} _ {sa} _ {sa} _ {sa} \ to \ mathscr {a} $ for \ mathscr {a} $，y Mathscr {a} $， $ \ mathscr {a} $。因此，我们称$ nc^k（\ mathbb {r}）$是非交换$ c^k $ functions的空间。我们的证明表明$ f_ \ Mathscr {a} \ in C^k（\ Mathscr {a} _ {sa}; \ Mathscr {a}）$，它仅需要对“多项运算符元素”的fréchet衍生物的知识（MOIS比较元素）（MOIS）（MOIS），是标准方法（MOIS），是标准方法，是标准方法的。尽管如此，$ nc^k（\ mathbb {r}）$包含所有可比较结果的功能。具体而言，我们证明$ nc^k（\ m athbb {r}）$包含同质的besov space $ \ dot {b} _1^{k，\ infty}（\ mathbb {r}但是，我们强调的是，本文的结果是其类型的第一个被证明用于任意Unital $ c^*$ - 代数，并且这种一般环境的扩展利用了作者最近解决MOI定义的某些“可分离性问题”的解决方案。最后，我们通过展示特定示例来证明$ W_K（\ Mathbb {r}）_ {loc} \ subsetNeq nc nc^k（\ m athbb {r}）\ subsetneq c^k（\ subsetneq c^k（\ supbbb {r}） $ k^{th} $ Wiener空间。
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 Fréchet 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 Fréchet 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 Hölder 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.
