A dichotomy for derivations and automorphisms of C*-algebras

Building on previous work of Kadison--Ringrose, Elliott, Akemann--Pedersen, and this author, we prove a dichotomy for the relation of outer equivalence of derivations and unitary equivalence of derivable automorphisms for a separable C*-algebra $A$: either such relations are trivial, or the relation $E_{0}^{\mathbb{N}}$ of tail equivalence of countably many binary sequences is reducible to them. When $A$ is furthermore \emph{unital}, this implies that $A$ has no outer derivation if and only if the group $\mathrm{Inn}\left( A\right)$ of inner automorphisms is $\boldsymbol{\Sigma }_{2}^{0}$ in $\mathrm{Aut}\left( A\right)$, if and only if it is $\boldsymbol{\Sigma }_{3}^{0}$ in $\mathrm{Aut}\left( A\right)$. Furthermore, one has that the space of inner derivations is norm-closed if and only if \textrm{Inn}$\left(A\right)$ is norm-closed, if and only if $\mathrm{Inn}\left( A\right)$ is $\boldsymbol{\Pi }_{3}^{0}$ in $\mathrm{\mathrm{Aut}}\left( A\right)$. This provides a complexity-theoretic characterization of C*-algebras with only inner derivations, which as a by-product rules out $D(\boldsymbol{\Pi }_{2}^{0})$ as a possible complexity class for $\mathrm{Inn}\left( A\right)$ in $\mathrm{\mathrm{Aut}}\left( A\right)$ for a separable unital C*-algebra $A$.