观察结果I：石头光谱
Observables I: Stone Spectra
摘要：
在这项工作中，我们讨论的概念可观察-量子和经典-从一个新的角度来看。在经典力学中，一个可观测量被表示为一个函数（可测量的、连续的或光滑的），而在量子物理学中，一个可观测量被表示为希尔伯特空间上的一个键合自伴算子。我们将在本书的第二部分说明，这两个不同概念背后有一个共同的结构。如果$\mathcal{R}$是冯诺依曼代数，则自伴元$A \in \mathcal{R}$诱导定义在$\mathcal{R}$中投影格$\mathcal{P(R)}$的\emph{Stone spectrum} $\mathcal{Q}(\mathcal{P(R)})$上的连续函数$f_{A} : \mathcal{Q}(\mathcal{P(R)}) \to \mathbb{R}$。一般格$\mathbb{L}$的Stone谱$\mathcal{Q}(\mathbb{L})$是$\mathbb{L}$中的极大对偶理想的集合，具有一个标准拓扑。如果$\mathbb{L}$是一个布尔代数（因此是“Stone”），并且在$\mathbb{L} = \mathcal{P(R)}$的情况下同胚于阿贝尔冯诺依曼代数$\mathcal{R}$的Gelfand谱（因此是“spectrum”），则$\mathcal{Q}(\mathbb{L})$与Stone的构造一致。
英文摘要：
In this work we discuss the notion of observable - both quantum and classical - from a new point of view. In classical mechanics, an observable is represented as a function (measurable, continuous or smooth), whereas in (von Neumann's approach to) quantum physics, an observable is represented as a bonded selfadjoint operator on Hilbert space. We will show in part II of this work that there is a common structure behind these two different concepts. If $\mathcal{R}$ is a von Neumann algebra, a selfadjoint element $A \in \mathcal{R}$ induces a continuous function $f_{A} : \mathcal{Q}(\mathcal{P(R)}) \to \mathbb{R}$ defined on the \emph{Stone spectrum} $\mathcal{Q}(\mathcal{P(R)})$ of the lattice $\mathcal{P(R)}$ of projections in $\mathcal{R}$. The Stone spectrum $\mathcal{Q}(\mathbb{L})$ of a general lattice $\mathbb{L}$ is the set of maximal dual ideals in $\mathbb{L}$, equipped with a canonical topology. $\mathcal{Q}(\mathbb{L})$ coincides with Stone's construction if $\mathbb{L}$ is a Boolean algebra (thereby ``Stone'') and is homeomorphic to the Gelfand spectrum of an abelian von Neumann algebra $\mathcal{R}$ in case of $\mathbb{L} = \mathcal{P(R)}$ (thereby ``spectrum'').
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