厄米完全有界映射的极小Wittstock包络
Minimal Wittstock Envelopes of Hermitian Completely Bounded Maps
- University of Bojnord（博伊努尔德大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究厄米完全有界映射的Wittstock包络，证明每个包络下有极小包络但不唯一，给出最小优超存在判据，并定义包络相对透视以推广Okayasu结果。
AI中文摘要：
厄米完全有界映射 $\Phi:\mathscr{A}\to B(\mathcal{H})$ 未必具有绝对值的典范完全正类比。我们通过 $\Phi$ 的完全正对称优超映射的有序集 $\mathsf{E}(\Phi)$（称为Wittstock包络）来研究此问题。我们证明每个Wittstock包络典范地确定其下方的一个极小Wittstock包络。然而，极小包络未必唯一。因此，我们研究更强的存在性问题：何时存在最小完全正优超映射。我们获得一个基于谱投影的中心支撑的判据，用于判定在固定包络之下最小优超映射的存在性。在有限维情形，Choi表示给出简单刻画：最小完全正优超映射存在当且仅当 $\Phi$ 或 $-\Phi$ 是完全正的。最后，我们利用包络相对模量定义厄米完全有界映射的包络相对透视。这些构造推广了Okayasu的完全正映射透视，在完全正情形退化为通常透视，并且一般依赖于所选的Wittstock包络。
英文摘要：
A Hermitian completely bounded map $Φ:\mathscr{A}\to B(\mathcal{H})$ need not have a canonical completely positive analogue of an absolute value. We study this problem through the ordered set $\mathsf{E}(Φ)$ of completely positive symmetric majorants of $Φ$, which we call Wittstock envelopes. We show that every Wittstock envelope canonically determines a minimal Wittstock envelope below it. However, minimal envelopes need not be unique. We therefore study the stronger question of when a least completely positive majorant exists. We obtain a criterion, in terms of central supports of spectral projections, for the existence of least majorants below a fixed envelope. In finite dimensions, the Choi representation gives a simple characterization: a least completely positive majorant exists if and only if \(Φ\) or \(-Φ\) is completely positive. Finally, we use the envelope-relative modulus to define envelope-relative perspectives for Hermitian completely bounded maps. These constructions extend Okayasu's completely positive map perspectives, reduce to the usual perspective in the completely positive case, and remain dependent in general on the chosen Wittstock envelopes.
