Derivation length and automorphism length of unital C*-algebras

Abstract

This paper is a contribution to the study of the ordinal-valued invariants of derivation and automorphism length. Akemann--Pedersen and Elliott proved that a separable unital C*-algebra has derivation length $0$ if and only if it has automorphism length at most $1/2$ if and only if it is a finite direct sum of homogeneous C*-algebras and simple C*-algebras. Kadison--Lance--Ringrose and Somerset proved that a separable unital C*-algebra has derivation length at most $1$ if and only if it has automorphism length at most $1$ if and only if its primitive spectrum has finite connecting order. In this paper, we prove a complete comparison between the two lengths: either \begin{equation*} \ell _{\mathrm{Aut}}\left( A\right) =\ell _{\mathrm{aut}}\left( A\right) \end{equation*} or, for some countable ordinal $\alpha $ other than $1$ or a limit ordinal, \begin{equation*} \ell _{\mathrm{aut}}\left( A\right) =\alpha \quad \text{and}\quad \ell _{\mathrm{Aut}}\left( A\right) =\alpha +1/2\text{.} \end{equation*}