## Abstract

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

-algebra L(Y) can be extended to a completely positive map φ

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 A to an injective C^{∗}^{∗}-algebra*L(Y)*as a linear combination of completely positive maps from A to*L(Y)*(always possible). We also discuss (ii) a noncommutative kernel generalization of the Arveson extension theorem (any completely positive map φ_{e}from a operator system S to an injective C^{∗}-algebra L(Y) can be extended to a completely positive map φ

_{e}from a C^{∗}-algebra containing S to*L(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).Original language | English |
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State | Published - 2 Feb 2022 |

## Keywords

- math.OA
- 47B32, 47A60