Tensor extension properties of C(K)-representations and applications to unconditionality

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.