# A dichotomy for derivations and automorphisms of C*-algebras

## TL;DR

This paper establishes a dichotomy in the complexity of relations between derivations and automorphisms of separable C*-algebras, linking their properties to descriptive set theory classifications.

## Contribution

It introduces a dichotomy for the relations of outer derivations and automorphisms, connecting algebraic properties with descriptive set theoretic complexity classifications.

## Key findings

- Outer derivations are trivial iff inner automorphisms are low complexity in automorphism group.
- The space of inner derivations is norm-closed iff inner automorphisms are low complexity.
- Provides a complexity-theoretic characterization of C*-algebras with only inner derivations.

## Abstract

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 $A$.