Nonsimplicity of certain universal C*-algebras

Abstract

Given $n\ge 2$, $z_{ij}\in \mathbb{T}$ such that $z_{ij}={\overline{z}}_{ji}$ for $1\le i,j\le n$ and $z_{ii}=1$ for $1\le i\le n$, and integers $p_1,\dots ,p_n\ge 1$, we show that the universal ${\mathrm{C}}^{*}$-algebra generated by unitaries $u_1,\dots ,u_n$ such that $u_i^{p_i}{u}_j^{p_j}=z_{ij}{u}_j^{p_j}{u}_i^{p_i}$ for $1\le i,j\le n$ 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.