Generalised Hecke algebras and C^*-completions

2006/09/14 by Landstad, Magnus B., Larsen, Nadia S. · 1 citation
#20C08 #46L55 #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.math/0609386

Abstract

For a Hecke pair (G, H) and a finite-dimensional representation σ of H on V_σ with finite range we consider a generalised Hecke algebra \H_σ(G, H), which we study by embedding the given Hecke pair in a Schlichting completion (G_σ, H_σ) that comes equipped with a continuous extension σ of H_σ. There is a (non-full) projection p_σ∈ C_c(G_σ, \cc B(V_σ)) such that \H_σ(G, H) is isomorphic to p_σC_c(G_σ, \cc B(V_σ))p_σ. We study the structure and properties of C^*-completions of the generalised Hecke algebra arising from this corner realisation, and via Morita-Fell-Rieffel equivalence we identify, in some cases explicitly, the resulting proper ideals of C^*(G_σ, \cc B(V_σ)). By letting σ vary, we can compare these ideals. The main focus is on the case with dimσ=1 and applications include ax+b-groups and the Heisenberg group.

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• Hecke C*-algebras, Schlichting completions, and Morita equivalence
• Hecke C*-algebras and semidirect products
• Quasi-symmetric group algebras and C*-completions of Hecke algebras
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• Geometric invariants of TDLC completions