Finitely generated nilpotent group C*-algebras have finite nuclear dimension

TL;DR

This paper proves that C*-algebras of finitely generated nilpotent groups have finite nuclear dimension, leading to classification results for certain irreducible representations satisfying the universal coefficient theorem.

Contribution

It establishes finite nuclear dimension for group C*-algebras of finitely generated nilpotent groups and classifies a broad class of their irreducible representations.

Findings

Group C*-algebras of finitely generated nilpotent groups have finite nuclear dimension.

Certain irreducible representations satisfy the universal coefficient theorem.

These representations are classifiable and approximately subhomogeneous.

Abstract

We show that group C*-algebras of finitely generated, nilpotent groups have finite nuclear dimension. It then follows, from a string of deep results, that the C*-algebra $A$ generated by an irreducible representation of such a group has decomposition rank at most 3. If, in addition, $A$ satisfies the universal coefficient theorem, another string of deep results shows it is classifiable by its Elliott invariant and is approximately subhomogeneous. We give a large class of irreducible representations of nilpotent groups (of arbitrarily large nilpotency class) that satisfy the universal coefficient theorem and therefore are classifiable and approximately subhomogeneous.