论文标题
免费的非交互性遗传内核：乔丹分解，阿尔维森扩展，内核统治
Free noncommutative hereditary kernels: Jordan decomposition, Arveson extension, kernel domination
论文作者
论文摘要
我们讨论了在紧凑的Hausdorff空间上进行复杂的Borel量度的（I）量化版本的（即，将一般非交换性核（标准函数标准概念的量化）分解为完全积极的非强制性核心（标准非官方kernel）的标准kernel（标准kernel）的标准kernel（标准kernel）标准kernel的一般性组合的更一般的问题。 （i）的其他特殊案例包括：分解一般操作符价值的内核作为正线性组合（并非总是可能）的线性组合，将一般有限的线性希尔伯特空间操作员分解为正线性操作员（始终可能）从$ c^$ c^$ alge $ c^*$ alge $ c^*$ alge的线性映射的线性组合（始终）的线性组合， $ c^*$ - algebra $ {\ mathcal l}（{\ Mathcal y}）$作为完全正面地图的线性组合，从$ {\ mathcal a} $到$ {\ sathcal l}（{\ Mathcal y}）$（始终可能）。 We also discuss (ii) a noncommutative kernel generalization of the Arveson extension theorem (any completely positive map $ϕ$ from a operator system ${\mathbb S}$ to an injective $C^*$-algebra ${\mathcal L}({\mathcal Y})$ can be extended to a completely positive map $ϕ_e$ 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).
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 $ϕ$ from a operator system ${\mathbb S}$ to an injective $C^*$-algebra ${\mathcal L}({\mathcal Y})$ can be extended to a completely positive map $ϕ_e$ 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).
