In general, commutative $C^\ast$-algebras are amenable. See [*A new proof of the amenability of $C(X)$*](http://dx.doi.org/10.1017/S0004972709001063) by Mortaza Abtahi and Yong Zhang (Bulletin of the Australian Mathematical Society, Volume 81, Issue 3, June 2010, pages 414-417).