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

Abstract

We show that for a Hecke pair (G,Γ ) the C * -completions C * (L 1 (G,Γ )) and pC * (G)p of its Hecke algebra coincide whenever the group algebra L 1 (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 [11]. Combining this result with our earlier results in [14] and a theorem of Tzanev [17] 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 [6] holds for all such Hecke pairs. We also show that the completions C * (L 1 (G,Γ )) and pC * (G)p do not always coincide, with the Hecke pair (SL 2 (Q q ), SL 2 (Z q )) providing one such example.

Positive definite *-spherical functions, property (T) and C*-completions of Gelfand pairs

“…Indeed, we prove the result in much greater generality. The proof differs from the case of n = 2 in [14] in that it appeals to property (T). After the second named author's talk on the case n = 2 at a NordForsk conference on the Farøe Islands in 2012, a discussion with M. Landstad revealed that the obstruction to injectivity in the case n = 3 would be property (T).…”

“…see [21] and [10]. This map is known to be an isomorphism in many cases, such as when G is Hermitian, [10], or when L 1 (G) is quasi-symmetric in the terminology of [14]. The last property is known to hold for a class of groups containing Hermitian groups and groups with subexponential growth.…”

“…However, to our knowledge, no proof of this claim has been published. The result showing that injectivity of the canonical map above fails for the pair (SL 2 (Q p ), SL 2 (Z p )) is due to the second named author, [14].…”

“…It is stated in [10, page 677] that the injectivity fails for the pair (PSL 3 (Q p ), PSL 3 (Z p )), according to a personal communication by Tzanev. However, to our knowledge, no proof of this claim has been published. The result showing that injectivity of the canonical map above fails for the pair (SL 2 (Q p ), SL 2 (Z p )) is due to the second named author, [14].…”

On the growth of Hermitian groups

“…These questions have already been asked by the author in [13], and the groups in questions 1) and 2) are natural to be considered in this regard. A negative answer to this question would imply that all Hermitian totally disconnected groups are amenable, which would give more evidence to the following long standing conjecture (see [14]):…”