C*-algebras generated by representations of virtually nilpotent groups

TL;DR

This paper proves that C*-algebras generated by irreducible representations of finitely generated virtually nilpotent groups satisfy key classification properties, including the universal coefficient theorem and real rank zero, extending classification results.

Contribution

It establishes that these C*-algebras are classified by their Elliott invariants and constructs explicit Cartan subalgebras in the nilpotent case, linking algebraic and representation-theoretic structures.

Findings

C*-algebras satisfy the universal coefficient theorem

C*-algebras have real rank zero

Explicit Cartan subalgebras are constructed for nilpotent groups

Abstract

We show that a C*-algebra generated by an irreducible representation of a finitely generated virtually nilpotent group satisfies the universal coefficient theorem and has real rank 0. This combines with previous joint work with Gillaspy and McKenney to show these C*-algebras are classified by their Elliott invariants. When we further assume the group is nilpotent we build explicit Cartan subalgebras that are closely related to the group and representation, although the Cartan subalgebras are generally not C*-diagonals.