  We discuss a (i) 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 $C^*$-algebra ${\mathcal A}$ to an injective $C^*$-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 $\phi$ from a operator system ${\mathbb
S}$ to an injective $C^*$-algebra ${\mathcal L}({\mathcal Y})$ can be extended
to a completely positive map $\phi_e$ from a $C^*$-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 positive).
