Abstract. We introduce completely bounded kernels taking values in ${\cal L}({\cal A}, {\cal B})$ where ${\cal A}$ and ${\cal B}$ are $C^*$-algebras. We show that if ${\cal B}$ is injective such kernels have a Kolmogorov decomposition precisely when they can be scaled to be completely contractive, and that this is automatic when the index set is countable.

AMS Subject Classification (1991): 46L07; 46L08, 46E22, 46B20

Keyword(s): completely bounded kernels, hermitian kernels, Kolmogorov decomposition

Received November 29, 2012, and in revised form December 14, 2012. (Registered under 103/2012.)