字典序泛函演算及其在泛函演算演算中的应用
Lexicographic functional calculus and its application to functional calculus calculus
AI总结：
本文提出字典序泛函演算（LFC）这一多元泛函演算，将其应用于泛函演算的微分演算，证明单变量泛函诱导映射的高阶Fréchet导数可由LFC表示，统一并推广了文献中关于泛函映射正则性的多类经典结果。
AI中文摘要：
设$A$为含单位元的$C^*$-代数，$I$为$A$的对称赋范理想。本文提出并研究字典序泛函演算（lexicographic functional calculus，简称LFC），这是一种针对$A$中非交换自伴元素组$(a_1,\ldots,a_m)$的多元泛函演算，这些元素“按字典序作用”，即从左到右作用，且在每个$i=1,\ldots,m-1$对应的$a_i$与$a_{i+1}$的作用之间“插入”一个元素$b_i \in I$。提出该演算的动因是其在“泛函演算演算”（即由单变量（连续）泛函演算诱导的映射的微分演算）中的应用。具体而言，本文证明：若$f\colon\mathbb{R}\to\mathbb{C}$具有足够正则性，且$a\in A_{\mathrm{sa}}:=\{c\in A:c^*=c\}$，则对所有$b\in I_{\mathrm{sa}}:=I\cap A_{\mathrm{sa}}$，有$f_{a,I}(b):=f(a+b)-f(a)\in I$；映射$f_{a,I}\colon I_{\mathrm{sa}}\to I$是Fréchet $C^k$的；且$f_{a,I}$的$k$阶Fréchet导数可通过作用于$f$的$k$阶均差（一个$k+1$元函数）的LFC表示。该结果恢复了或在极大程度上推广了文献中几乎所有同类结果。例如，它同时恢复了关于由$a\mapsto f(a)$定义的函数$f_A\colon A_{\mathrm{sa}}\to A$的正则性的以下三个高度相关的结果：(1) 若$A$是交换的且$f\in C^k(\mathbb{R})$，则$f_A$是Fréchet $C^k$的；(2) 若$A$是有限维的且$f\in C^k(\mathbb{R})$，则$f_A$是Fréchet $C^k$的；(3) 若$f\colon\mathbb{R}\to\mathbb{C}$的正则性“略优于$C^k$”，例如属于齐次Besov空间$\dot{B}_1^{k,\infty}(\mathbb{R})$，则无论$A$如何选取，$f_A$都是Fréchet $C^k$的。在LFC出现之前，不存在能统一结果(1)—(3)的框架；尤其不存在一个能将这三个结果全部作为推论的统一结论。
英文摘要：
Let $A$ be a unital $C^*$-algebra and $I$ be a symmetrically normed ideal of $A$. I introduce and study lexicographic functional calculus (LFC), a kind of multivariate functional calculus for tuples $(a_1,\ldots,a_m)$ of noncommuting self-adjoint elements of $A$ ''acting in lexicographic order,'' i.e., from left to right, with an element $b_i \in I$ ''inserted'' between the action of $a_i$ and $a_{i+1}$ for each $i=1,\ldots,m-1$. The reason for its introduction is an application to ''functional calculus calculus,'' the differential calculus of maps induced by single-variate (continuous) functional calculus. Specifically, I prove that if $f\colon\mathbb{R}\to\mathbb{C}$ is sufficiently regular and $a\in A_{\mathrm{sa}}:=\{c\in A:c^*=c\}$, then $f_{a,I}(b):=f(a+b)-f(a)\in I$ for all $b\in I_{\mathrm{sa}}:=I\cap A_{\mathrm{sa}}$, the map $f_{a,I}\colon I_{\mathrm{sa}}\to I$ is Fréchet $C^k$, and the $k^{\text{th}}$ Fréchet derivative of $f_{a,I}$ may be written in terms of LFC applied to the $k^{\text{th}}$ divided difference of $f$, a function of $k+1$ variables. This result recovers or vastly generalizes nearly all comparable results in the literature. For example, it simultaneously recovers the following three highly related results on the regularity of the function $f_A\colon A_{\mathrm{sa}}\to A$ defined by $a\mapsto f(a)$: (1) If $A$ is commutative and $f\in C^k(\mathbb{R})$, then $f_A$ is Fréchet $C^k$; (2) if $A$ is finite dimensional and $f\in C^k(\mathbb{R})$, then $f_A$ is Fréchet $C^k$; and (3) if $f\colon\mathbb{R}\to\mathbb{C}$ is ''slightly better than $C^k$,'' e.g., belongs to the homogeneous Besov space $\dot{B}_1^{k,\infty}(\mathbb{R})$, then $f_A$ is Fréchet $C^k$ no matter the choice of $A$. Before LFC, there was no unifying framework for results (1)--(3); in particular, there was no single result of which they were all corollaries.
