Illinois Journal of Mathematics

Simplicity of the reduced $C^{\ast}$-algebras of certain Coxeter groups

Gero Fendler

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Let $(G,S)$ be a finitely generated Coxeter group such that the Coxeter system is indecomposable and the canonical bilinear form is indefinite but non-degenerate. We show that the reduced $C^{\ast}$-algebra of $G$ is simple with unique normalised trace.

For an arbitrary finitely generated Coxeter group we prove the validity of a Haagerup inequality: There exist constants $C \gt 0$ and $\Lambda\in \mathbb{N}$ such that, for any function $f\in l^2(G)$ supported on elements of length $n$ with respect to the generating set $S$ and for all $h\in l^2(G)$, $\|\,f\ast h\,\| \leq C(n+1)^{\frac{3}{2}{\Lambda}}\|\,f\,\|$.

Article information

Illinois J. Math., Volume 47, Number 3 (2003), 883-897.

First available in Project Euclid: 13 November 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46L05: General theory of $C^*$-algebras
Secondary: 20F55: Reflection and Coxeter groups [See also 22E40, 51F15] 22D15: Group algebras of locally compact groups 46L55: Noncommutative dynamical systems [See also 28Dxx, 37Kxx, 37Lxx, 54H20]


Fendler, Gero. Simplicity of the reduced $C^{\ast}$-algebras of certain Coxeter groups. Illinois J. Math. 47 (2003), no. 3, 883--897. doi:10.1215/ijm/1258138199.

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