Quasi-symmetric Group Algebras and C ^∗-Completions of Hecke Algebras

Abstract

We show that for a Hecke pair (G, Γ) the C ^∗-completions ${C}^{{\ast}}({L}^{1}(G,\varGamma ))$ and $p{C}^{{\ast}}(\overline{G})p$ of its Hecke algebra coincide whenever the group algebra ${L}^{1}(\overline{G})$ 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 et al. (Proc Edinb Math Soc (2) 51(3):657–695, 2008). Combining this result with our earlier results in (Palma, J Funct Anal 264:2704–2731, 2013) and a theorem of Tzanev (J Oper Theory 50(1):169–178, 2003) 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 a nilpotent group. As a consequence, the category equivalence studied by Hall (Hecke C ^∗-algebras. Ph.D. thesis, The Pennsylvania State University, 1999) holds for all such Hecke pairs. We also show that the completions ${C}^{{\ast}}({L}^{1}(G,\varGamma ))$ and $p{C}^{{\ast}}(\overline{G})p$ do not always coincide, with the Hecke pair $(SL_{2}(\mathbb{Q}_{q}),SL_{2}(\mathbb{Z}_{q}))$ providing one such example.

Mathematics Subject Classification (2010): 46L55, 20C08.