Completely bounded kernels

Details Completely bounded kernels Completely bounded kernels

2010-01-20

arxiv.org/abs/1001.3590v2

Mathematics

Operator Algebras

18 pages. This version corrects a gap in the proof of Theorem 6.1

Scientific paper

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.

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