Rings of Bounded Continuous Functions

TL;DR

This paper explores the algebraic structure of rings of bounded continuous functions, establishing dualities with topological spaces and providing new proofs and methods for classical theorems in topology and functional analysis.

Contribution

It introduces a unified algebraic framework for BC R-rings and BC C-rings, offering new proofs and potential progress on classical topological concepts.

Findings

Category of BC R-rings is dual to compact topological spaces

Ring of bounded continuous functions forms a Banach K-ring

Characterization of Stone spaces via BC rings

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.