Classification of C*-algebras generated by representations of the unitriangular group $UT(4,\mathbb{Z})$

TL;DR

This paper computes the ordered K-theory of C*-algebras generated by faithful irreducible representations of the three-step nilpotent group $UT(4, obreak \\mathbb{Z})$, showing they are all simple A$ obreak \\mathbb{T}$ algebras and highlighting the existence of many non-A$ obreak \\mathbb{T}$ simple algebras from nilpotent groups.

Contribution

It explicitly calculates the ordered K-theory for these C*-algebras and demonstrates their classification as simple A$ obreak \\mathbb{T}$ algebras, expanding understanding of their structure.

Findings

All such C*-algebras are simple A$ obreak \\mathbb{T}$ algebras.

Many simple non-A$ obreak \\mathbb{T}$ algebras also arise from nilpotent groups.

Ordered K-theory fully classifies these C*-algebras.

Abstract

It was recently shown that each C*-algebra generated by a faithful irreducible representation of a finitely generated, torsion free nilpotent group is classified by its ordered K-theory. For the three step nilpotent group $UT(4,\mathbb{Z})$ we calculate the ordered K-theory of each C*-algebra generated by a faithful irreducible representation of $UT(4,\mathbb{Z})$ and see that they are all simple A$\mathbb{T}$ algebras. We also point out that there are many simple non A$\mathbb{T}$ algebras generated by irreducible representations of nilpotent groups.