Nonsimplicity of certain universal C^*-algebras

Abstract

Given n≥2" role="presentation">n≥2, zij∈T" role="presentation">zij∈T such that zij=z¯ji" role="presentation">zij=z¯ji for 1≤i,j≤n" role="presentation">1≤i,j≤n and zii=1" role="presentation">zii=1 for 1≤i≤n" role="presentation">1≤i≤n, and integers p1,…,pn≥1" role="presentation">p1,…,pn≥1, we show that the universal C*" role="presentation">C∗-algebra generated by unitaries u1,…,un" role="presentation">u1,…,un such that uipiujpj=zijujpjuipi" role="presentation">upiiupjj=zijupjjupii for 1≤i,j≤n" role="presentation">1≤i,j≤n is not simple if at least one exponent pi" role="presentation">pi is at least two. We indicate how the method of proof by “working with various quotients” can be used to establish nonsimplicity of universal C*" role="presentation">C∗-algebras in other cases.