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

Christoph Kriegler (1) , Christian Le Merdy (1)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

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.

Mots clés: R-boundedness; Unconditionality; functional calculus

Domaines: Analyse fonctionnelle [math.FA]

Christoph Kriegler, Christian Le Merdy. Tensor extension properties of C(K)-representations and applications to unconditionality. Journal of the Australian Mathematical Society, 2010, 88, pp.205-230. ⟨hal-00477656⟩