论文标题
非交通性几何和操作员系统中的光谱截断
Spectral truncations in noncommutative geometry and operator systems
论文作者
论文摘要
在本文中，我们扩展了传统的非共同几何形状框架，以应对几何空间的光谱截断（即在动量空间中施加紫外线截止），并具有耐受性关系，从而提供有限分辨率的几何空间的粗粒近似值。在我们的新方法中，$ c^*$ - 代数由操作员系统接管。作为技术的一部分，我们将$ c^*$ - 信封，双操作员系统和稳定的等效性。我们为运算符系统定义了一个传播数，在稳定的等效性下，我们表明它是一个不变的，用于比较同一空间的近似值。 我们说明了通过圆的光谱截断获得的混凝土示例的方法。这些是有限维toeplitz矩阵及其双重操作员系统的运算符系统，它们由整数组的组代数中的函数在固定的间隔内支持。事实证明，这些操作员系统的正元素和纯状态空间具有非常丰富的结构，我们分析了，包括正锥的边界的代数几何和度量方面的代数几何形状，即与DIRAC操作员相关的状态空间的距离。光谱截断的主要特性是它使等轴测组完整。相反，如果一个人考虑了循环矩阵提供的另一个有限近似，则等轴测组将离散，即使在这种情况下，操作员系统为$ c^*$ - 代数。我们在有限的傅立叶变换的背景下对此进行了分析。 将非交通性几何形状扩展到运算符系统允许一个人通过考虑两个点之间的关系$ d（x，y）<ε$在两个点之间或更一般而言的公差关系之间来处理有限分辨率的度量空间，从而自然会引起操作员系统。
In this paper we extend the traditional framework of noncommutative geometry in order to deal with spectral truncations of geometric spaces (i.e. imposing an ultraviolet cutoff in momentum space) and with tolerance relations which provide a coarse grain approximation of geometric spaces at a finite resolution. In our new approach the traditional role played by $C^*$-algebras is taken over by operator systems. As part of the techniques we treat $C^*$-envelopes, dual operator systems and stable equivalence. We define a propagation number for operator systems, which we show to be an invariant under stable equivalence and use to compare approximations of the same space. We illustrate our methods for concrete examples obtained by spectral truncations of the circle. These are operator systems of finite-dimensional Toeplitz matrices and their dual operator systems which are given by functions in the group algebra on the integers with support in a fixed interval. It turns out that the cones of positive elements and the pure state spaces for these operator systems possess a very rich structure which we analyze including for the algebraic geometry of the boundary of the positive cone and the metric aspect i.e. the distance on the state space associated to the Dirac operator. The main property of the spectral truncation is that it keeps the isometry group intact. In contrast, if one considers the other finite approximation provided by circulant matrices the isometry group becomes discrete, even though in this case the operator system is a $C^*$-algebra. We analyze this in the context of the finite Fourier transform. The extension of noncommutative geometry to operator systems allows one to deal with metric spaces up to finite resolution by considering the relation $d(x,y)<ε$ between two points, or more generally a tolerance relation which naturally gives rise to an operator system.
