We give a sufficient condition for a * -algebra with a specified basis to have an enveloping C * -algebra. Particularizing to the setting of a Hecke algebra H(G, Γ), we show that under a suitable assumption not only we can assure that an enveloping C * -algebra C * (G, Γ) exists, but also that it coincides with C * (L 1 (G, Γ)), the enveloping C * -algebra of the L 1 -Hecke algebra. Our methods are used to show the existence of C * (G, Γ) and isomorphism with C * (L 1 (G, Γ)) for several classes of Hecke algebras. Most of the classes which are known to satisfy these properties are covered by this approach, and we also describe some new ones.

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.

We develop a theory of crossed products by "actions" of Hecke pairs (G, Γ), motivated by applications in non-abelian C * -duality. Our approach gives back the usual crossed product construction whenever G/Γ is a group and retains many of the aspects of crossed products by groups. In this first of two articles we lay the * -algebraic foundations of these crossed products by Hecke pairs and we explore their representation theory.

The study of existence of a universal C * -completion of the * -algebra canonically associated to a Hecke pair was initiated by Hall, who proved that the Hecke algebra associated to (SL2(Qp), SL2(Zp)) does not admit a universal C * -completion. Kaliszewski, Landstad and Quigg studied the problem by placing it in the framework of Fell-Rieffel equivalence, and highlighted the role of other C * -completions. In the case of the pair (SLn(Qp), SLn(Zp)) for n ≥ 3 we show, invoking property (T) of SLn(Qp), that the C * -completion of the L 1 -Banach algebra and the corner of C * (SLn(Qp)) determined by the subgroup are distinct. In fact, we prove a more general result valid for a simple algebraic group of rank at least 2 over a p-adic field with a good choice of a maximal compact open subgroup.

A locally compact group G is said to be Hermitian if every selfadjoint element of L 1 (G) has real spectrum. Using Halmos' notion of capacity in Banach algebras and a result of Jenkins, Fountain, Ramsay and Williamson we will put a bound on the growth of Hermitian groups. In other words, we will show that if G has a subset that grows faster than a certain constant, then G cannot be Hermitian. Our result allows us to give new examples of non-Hermitian groups which could not tackled by the existing theory. The examples include certain infinite free Burnside groups, automorphism groups of trees, and p-adic general and special linear groups.