$C^*$-algebras with finite complexity

TL;DR

This paper explores the concept of complexity rank in $C^*$-algebras, introduces a weak variant, and characterizes their ranks in relation to properties like nuclear dimension, real rank, and $K$-theory, with implications for the UCT.

Contribution

It introduces the weak complexity rank, establishes its equivalence with known properties in certain cases, and computes the complexity rank for all UCT Kirchberg algebras.

Findings

Weak complexity rank one equals nuclear dimension one and real rank zero.

Complexity rank of UCT Kirchberg algebras is either one or two.

Weak complexity rank and complexity rank differ due to $K$-theoretic obstructions.

Abstract

Complexity rank for $C^*$-algebras was introduced by the second author and Yu for applications towards the UCT: very roughly, this rank is at most $n$ if you can repeatedly cut the $C^*$-algebra in half at most $n$ times, and end up with something finite dimensional. In this paper, we study complexity rank, and also a weak complexity rank that we introduce; having weak complexity rank at most one can be thought of as `two-colored local finite-dimensionality'. We first show that for separable, unital, and simple $C^*$-algebras, weak complexity rank one is equivalent to the conjunction of nuclear dimension one and real rank zero. In particular, this shows that the UCT for all nuclear $C^*$-algebras is equivalent to equality of the weak complexity rank and the complexity ranks for Kirchberg algebras with zero $K$-theory groups. However, we also show using a $K$-theoretic obstruction (torsion in $K_1$) that weak complexity rank one and complexity rank one are not the same in general. We then use the Kirchberg-Phillips classification theorem to compute the complexity rank of all UCT Kirchberg algebras: it is always one or two, with the rank one case occurring if and only if the $K_1$-group is torsion free.