论文标题
有限连续功能的环
Rings of Bounded Continuous Functions
论文作者
论文摘要
我们使用交换代数的方法从拓扑和功能分析中检查了几种经典概念。我们表明，这些各种概念都由BC R形环及其最大光谱控制。 BC R环是在某些紧凑的拓扑空间x上与有限连续的R值函数同构的环A。这些环未拓扑。 我们证明，BC R形环的类别是紧凑型拓扑空间的类别。接下来，我们证明，对于每个拓扑空间x，它上有界连续函数的环是BC R形环。这些定理合并得出了任意拓扑空间的石技术压缩的代数结构。 BC C形环有类似的概念。每个BC C-RING A都有规范的互动。 A的规范遗产分子是BC R形环，这是从BC C形环到BC R形环的类别的等效性。 令K为R或C。我们证明BC K-RING A上有规范的规范，使其成为Banach K-Ring。然后，我们证明健忘的函子是从Banach^* k-rings（更称为交换性的Unital C^* k-algebras）到BC K-Ring的等价性。健忘函子的准插曲赋予了BC K-Ring的规范，并在k = C时的规范差异。 石材拓扑空间，也称为涂鸦拓扑空间，传统上与布尔环相关 - 这是石材二元性。我们给出石头空间的BC环特征。由此，我们获得了一个非常简单的证明，证明了离散空间的石技术压实是石材空间。 本文中的大多数结果并不是什么新鲜事。但是，我们的大多数证据似乎是新的 - 我们的方法可能会导致与这些古典主题有关的真正进步。
We examine several classical concepts from topology and functional analysis, using methods of commutative algebra. We show that these various concepts are all controlled by BC R-rings and their maximal spectra. A BC R-ring is a ring A that is isomorphic to the ring of bounded continuous R-valued functions on some compact topological space X. These rings are not topologized. We prove that the category of BC R-rings is dual to the category of compact topological spaces. Next we prove that for every topological space X the ring of bounded continuous functions on it is a BC R-ring. These theorems combined yield an algebraic construction of the Stone-Cech Compactification of an arbitrary topological space. There is a similar notion of BC C-ring. Every BC C-ring A has a canonical involution. The canonical hermitian subring of A is a BC R-ring, and this is an equivalence of categories from BC C-rings to BC R-rings. Let K be either R or C. We prove that a BC K-ring A has a canonical norm on it, making it into a Banach K-ring. We then prove that the forgetful functor is an equivalence from Banach^* K-rings (better known as commutative unital C^* K-algebras) to BC K-rings. The quasi-inverse of the forgetful functor endows a BC K-ring with its canonical norm, and the canonical involution when K = C. Stone topological spaces, also known as profinite topological spaces, are traditionally related to boolean rings - this is Stone Duality. We give a BC ring characterization of Stone spaces. From that we obtain a very easy proof of the fact that the Stone-Cech Compactification of a discrete space is a Stone space. Most of the results in this paper are not new. However, most of our proofs seem to be new - and our methods could potentially lead to genuine progress related to these classical topics.
