Nonsimplicity of certain universal \(C^{\ast}\)-algebras

Abstract: Given

n g e q 2

,

z_{i j} i n m a t h b b T

such that

z_{i j} = o v e r l i n e z_{j i}

for

1 l e q i, j l e q n

and

z_{i i} = 1

for

1 l e q i l e q n

, and integers

p_{1}, . . ., p_{n} g e q 1

, we show that the universal

m a t h r m C^{*}

-algebra generated by unitaries

u_{1}, . . ., u_{n}

such that

u_{i}^{p_{i}} u_{j}^{p_{j}} = z_{i j} u_{j}^{p_{j}} u_{i}^{p_{i}}

for

1 l e q i, j l e q 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

m a t h r m C^{*}

-algebras in other cases.

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