论文标题
旋转单位的完全订单等效性
Complete Order Equivalence of Spin Unitaries
论文作者
论文摘要
本文是对矩阵代数上矩阵和线性图的线性空间的研究，这些矩阵代数是由\ emph {spin systems}或\ emph {spin nimaries}引起的，这些有限的设置是$ \ mathcal s $ selfadchoint单位矩阵的$ \ mathcal s $ s $ selfrices nitrices，以便在$ \ mathcal s Anticalcal的任何两个单位者中均属于$ \ nimaries。我们对矩阵的这些线性空间之间的线性同构特别感兴趣，以便保留这些空间内的矩阵秩序。这种同构被称为完全秩序同构，这可能被视为单一相似性的较弱概念。本文的主要结果表明，从这个意义上讲，所有$ m $ $ $ thus的抗跨性别偶像统一矩阵都是等效的，这意味着这些元素会产生的Unital线性子空间之间存在一个Unital Complete Order Order同构。我们还表明，由基数$ 2K $或$ 2K+1 $生成的任何运算符系统的C $^*$ - 简单矩阵algebra $ \ mathcal m_ {2^k}（\ Mathbb c）$。作为主要结果的应用，我们表明，由自旋单位型确定的自由谱仅取决于单位的数量，而不是特定的单位选择，我们给出了事实的新直接证明[Helton-Klep-McCullogh-McCullough-Schweighofer，“ linations”，linear matrix Inorix Inorix Inorix Inorix Inorix cobe and Matrix Cube问题，beta beta beta bema bema bemabe'bem bem bem bem bem bem bem bem bem。阿米尔。数学。 Soc。 257（2019）]旋转球$ \ mathfrak b_m^{\ rm spin} $ and max ball $ \ mathfrak b_m^{\ rm max} $ comincide as Concide as as astrix sucex seets $ m = 1,2 $。我们还获得了可计数旋转系统及其C $^*$信封的类似结果。
This paper is a study of linear spaces of matrices and linear maps on matrix algebras that arise from \emph{spin systems}, or \emph{spin unitaries}, which are finite sets $\mathcal S$ of selfadjoint unitary matrices such that any two unitaries in $\mathcal S$ anticommute. We are especially interested in linear isomorphisms between these linear spaces of matrices such that the matricial order within these spaces is preserved; such isomorphisms are called complete order isomorphisms, which might be viewed as weaker notion of unitary similarity. The main result of this paper shows that all $m$-tuples of anticommuting selfadjoint unitary matrices are equivalent in this sense, meaning that there exists a unital complete order isomorphism between the unital linear subspaces that these tuples generate. We also show that the C$^*$-envelope of any operator system generated by a spin system of cardinality $2k$ or $2k+1$ is the simple matrix algebra $\mathcal M_{2^k}(\mathbb C)$. As an application of the main result, we show that the free spectrahedra determined by spin unitaries depend only upon the number of the unitaries, not upon the particular choice of unitaries, and we give a new, direct proof of the fact [Helton-Klep-McCullough-Schweighofer, "Dilations, linear matrix inequalities, the matrix cube problem and beta distributions", Mem. Amer. Math. Soc. 257(2019)] that the spin ball $\mathfrak B_m^{\rm spin}$ and max ball $\mathfrak B_m^{\rm max}$ coincide as matrix convex sets in the cases $m=1,2$. We also derive analogous results for countable spin systems and their C$^*$-envelopes.
