Free noncommutative hereditary kernels: Jordan decomposition, Arveson extension, kernel domination

Abstract

We discuss (i) a quantized version of the Jordan decomposition theorem for a complex Borel measure on a compact Hausdorff space, namely, the more general problem of decomposing a general noncommutative kernel (a quantization of the standard notion of kernel function) as a linear combination of completely positive noncommutative kernels (a quantization of the standard notion of positive definite kernel). Other special cases of (i) include: the problem of decomposing a general operator-valued kernel function as a linear combination of positive kernels (not always possible), of decomposing a general bounded linear Hilbert-space operator as a linear combination of positive linear operators (always possible), of decomposing a completely bounded linear map from a $C^{\ast }$-algebra $\mathcal{A}$ to an injective $C^{\ast }$-algebra $\mathcal{L}(\mathcal{Y})$ as a linear combination of completely positive maps from $\mathcal{A}$ to $\mathcal{L}(\mathcal{Y})$ (always possible). We also discuss (ii) a noncommutative kernel generalization of the Arveson extension theorem (any completely positive map $\varphi$ from an operator system $\mathbb{S}$ to an injective $C^{\ast }$-algebra $\mathcal{L}(\mathcal{Y})$ can be extended to a completely positive map ${\varphi }_e$ from a $C^{\ast }$-algebra containing $\mathbb{S}$ to $\mathcal{L}(\mathcal{Y}))$, and (iii) a noncommutative kernel version of a Positivstellensatz (i.e., finding a certificate to explain why one kernel is positive at points where another given kernel is strictly positive).