On enveloping C⁎-algebras of Hecke algebras

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⁎(L1(G,Γ)), the enveloping C⁎-algebra of the L1-Hecke algebra. Our methods are used to show the existence of C⁎(G,Γ) and isomorphism with C⁎(L1(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.