Applications of the Gelfand--Naimark duality

TL;DR

This paper explores the applications of the Gelfand--Naimark duality, highlighting its insights into compact Hausdorff spaces, *-algebras, and related topological structures, extending classical dualities like Stone and Wallman.

Contribution

It demonstrates how Gelfand--Naimark duality offers new perspectives on compact spaces and their *-algebraic representations, especially for ech--Stone remainders.

Findings

Provides a detailed analysis of Gelfand--Naimark duality in topological contexts

Connects duality theory with *-algebraic structures of compact spaces

Highlights implications for autohomeomorphisms of ech--Stone remainders

Abstract

Stone duality is an indispensable tool for the study of compact, zero-dimensional, Hausdorff spaces. In the case of general compact Hausdorff spaces one can get quite a bit of mileage by considering the `Wallman duality' between compact spaces and lattices of closed sets. I will argue that the Gelfand--Naimark duality between compact Hausdorff spaces and unital, commutative \cstar-algebras provides great insight into compact Hausdorff spaces, and \v Cech--Stone remainders and their autohomeomorphisms in particular.