Let $K$ be any compact set. The $C^*$-algebra $C(K)$ is nuclear and any bounded homomorphism from $C(K)$ into $B(H)$, the algebra of all bounded operators on some Hilbert space $H$, is automatically completely bounded. We prove extensions of these results to the Banach space setting, using the key concept of $R$-boundedness. Then we apply these results to operators with a uniformly bounded $\HI$-calculus, as well as to unconditionality on $L^p$. We show that any unconditional basis on $L^p$ `is' an unconditional basis on $L^2$ after an appropriate change of density.