  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 $\h(G,\gm)$, we show that under a suitable assumption not only we can
assure that an enveloping $C^*$-algebra $C^*(G, \gm)$ exists, but also that it
coincides with $C^*(L^1(G,\gm))$, the enveloping $C^*$-algebra of the
$L^1$-Hecke algebra. Our methods are used to show the existence of $C^*(G,
\gm)$ and isomorphism with $C^*(L^1(G,\gm))$ 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.
