Lexicographic functional calculus and its application to functional calculus calculus

Abstract

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.