Rings of Bounded Continuous Functions (2202.08334v2)

Abstract: 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.