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

TL;DR

This paper extends classical properties of C(K)-representations to Banach spaces using R-boundedness, and applies these results to operators with bounded H^-calculus and unconditional bases on L^p spaces.

Contribution

It introduces tensor extension properties for C(K)-representations in Banach spaces and demonstrates their applications to operator calculus and unconditionality in L^p spaces.

Findings

Bounded homomorphisms from C(K) are completely bounded in the Banach space setting.

Operators with bounded H^-calculus are analyzed using these extension properties.

Unconditional bases on L^p are shown to correspond to those on L^2 after a change of density.

Abstract

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.