  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 H^\infty-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.
