Completely bounded kernels

TL;DR

This paper introduces a new class of kernels called completely bounded kernels valued in operator spaces between C*-algebras, establishing conditions for their decomposition and automatic properties in countable cases.

Contribution

It defines completely bounded kernels in the context of C*-algebras and characterizes their decompositions and automatic contractiveness under certain conditions.

Findings

Kernels have a Kolmogorov decomposition when scaled to be completely contractive.

Automatic decomposition occurs when the index set is countable.

The results connect kernel properties with the structure of C*-algebras.

Abstract

We introduce completely bounded kernels taking values in L(A,B) where A and B are C*-algebras. We show that if 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.