Analysis Seminar: Lexicographic functional calculus

Abstract: For a unital $C^*$-algebra $A$ and a continuous function $f \colon \mathbb{R} \to \mathbb{C}$, let $f_A \colon A_{\mathrm{sa}} \to A$ be the map taking a self-adjoint element $a \in A$ to the (normal) element $f(a) \in A$ defined via the continuous functional calculus.
It is easy to show that $f_A$ is continuous, so it is natural to wonder whether $f \in C^k(\mathbb{R})$ implies $f_A \in C^k(A_{\mathrm{sa}};A)$ whenever $k \in \mathbb{N}$.  Interestingly, this is not generally true.  The state of the art is that (1) if $f$ ``is slightly better than $C^k$,'' e.g., belongs to the H\"older space $C_{\mathrm{loc}}^{k,\varepsilon}(\R)$ or the Besov space $B_1^{k,\infty}(\R)$, then $f_A$ is $C^k$, no matter the choice of $A$.  However, if $A$ is (2) commutative or (3) finite dimensional, then $f \in C^k(\R)$ actually \emph{does} imply $f_A \in C^k(A_{\mathrm{sa}};A)$.  Though the proofs of (1)--(3) are related, there is presently no general result of which these results are corollaries.  I propose a new framework for a kind of functional calculus, called \emph{lexicographic functional calculus}, for noncommuting tuples of self-adjoint elements of $A$ that enables the formulation and proof of such a general result.

Host: John McCarthy