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.
本文填补了作者的论文“有界元的迭代环与Schmüdgen定理的推广”[1]和作者的文章“有界元的迭代环与Schmüdgen正性定理的推广”[2]中同时出现的一个空白。[1]康斯坦茨大学（2001年），可在http://www.ub.uni-konstanz.de/kops/volltexte/2002/825/上查阅 [2] Journal für die reine und angewandte Mathematik 554，19-45（2003年）
几年前，Bondal，Rouquier和Van den Bergh引入了三角范畴维数的概念，Rouquier证明了完满域上有限型分离有限型概形上凝聚层的有界导出范畴具有有限维数.本文研究交换诺特环上有限生成模的有界导出范畴的维数。本文的主要结果是：它在包含具有完全剩余域的域的完全局部环上是有限的。我们的方法也给出了一个环理论证明的仿射的情况下，Rouquier定理。
