Simplicity of the Reduced C-algebras of Certain Coxeter Groups

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∗-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 > 0 and Λ ∈ N such that for a function f ∈ l(G) supported on elements of length n with respect to the generating set S: ‖ f ∗ h ‖2 ≤ C(n + 1) 3 2 ‖ f ‖2‖h ‖2, ∀h ∈ l (G). 0AMS Mathematical Subject Classification(2000): primary: 43A65; secondary: 46L99, 22D25, 20F55