It is a classical result that scalar valued positive kernels have Kolmogorov
decompositions. This has been extended in various ways, culminating in a
version of the Kolmogorov decomposition for completely positive L(A,B) valued
kernels, A and B C*-algebras, due to Barreto, Bhat, Liebscher and Skeide. The
notion of a Kolmogorov decomposition has also been extended to operator valued
hermitian, though not necessarily positive, kernels by Constantinescu and
Gheondea building on work of Schwartz, where a condition for decomposability is
shown to be that the kernel can be written as a difference of positive kernels.
For L(A,B) valued kernels, the appropriate analogue is that of a completely
bounded kernel, which we define in both the hermitian and non-hermitian case.
We show that the so-called Schwartz boundedness condition implies the existence
of a Kolmogorov decomposition for Hermitian kernels, and that when A is unital
and B is injective (much as in the Wittstock decomposition theorem), completely
bounded kernels have Kolmogorov decompositions.