2012/10/14 by Rui Palma, Palma, Rui
Mathematics · #Advanced Operator Algebra Research #Advanced Topics in Algebra #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.1210.3807
We show that for a Hecke pair (G, Γ) the C^*-completions C^*(L1(G, Γ)) and pC^*(G)p of its Hecke algebra coincide whenever the group algebra L1(G) satisfies a spectral property which we call "quasi-symmetry", a property that is satisfied by all Hermitian groups and all groups with subexponential growth. We generalize in this way a result of Kaliszewski, Landstad and Quigg. Combining this result with our earlier results and a theorem of Tzanev we establish that the full Hecke C^*-algebra exists and coincides with the reduced one for several classes of Hecke pairs, particularly all Hecke pairs (G, Γ) where G is nilpotent group. As a consequence, the category equivalence studied by Hall holds for all such Hecke pairs. We also show that the completions C^*(L1(G, Γ)) and pC^*(G)p do not always coincide, with the Hecke pair (SL2(ℚq), SL2(ℤq)) providing one such example.