Nonsimplicity of certain universal $\mathrm{C}^\ast$-algebras

TL;DR

This paper proves that certain universal C*-algebras generated by unitaries with specified commutation relations are not simple when at least one generator's power is greater than one, using quotient methods.

Contribution

It demonstrates the nonsimplicity of a class of universal C*-algebras under specific conditions, introducing a quotient-based proof technique.

Findings

Universal C*-algebras are not simple if any exponent p_i ≥ 2.

Method of working with quotients can establish nonsimplicity in other cases.

Provides insight into the structure of C*-algebras with specific commutation relations.

Abstract

Given $n\geq 2$, $z_{ij}\in\mathbb{T}$ such that $z_{ij}=\overline z_{ji}$ for $1\leq i,j\leq n$ and $z_{ii}=1$ for $1\leq i\leq n$, and integers $p_1,...,p_n\geq 1$, we show that the universal $\mathrm{C}^*$-algebra generated by unitaries $u_1,...,u_n$ such that $u_i^{p_i}u_j^{p_j}=z_{ij}u_j^{p_j}u_i^{p_i}$ for $1\leq i,j \leq n$ is not simple if at least one exponent $p_i$ is at least two. We indicate how the method of proof by `working with various quotients' can be used to establish nonsimplicity of universal $\mathrm{C}^*$-algebras in other cases.