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.