Nonsimplicity of certain universal ${\mathrm{C}}^{\ast }$-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 $p_1,...,p_n\ge 1$, we show that the universal ${\mathrm{C}}^{\ast }$-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 $1\le i,j\le n$ 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 ${\mathrm{C}}^{\ast }$-algebras in other cases.

3 pages. Small improvements from the first version. Final version, to appear in Annals of Functional Analysis

References in corpus (1)

• Every simple higher dimensional noncommutative torus is an AT algebra