Applications of the Gelfand--Naimark duality

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 C*-algebras provides great insight into compact Hausdorff spaces as well as into \v Cech--Stone remainders and their autohomeomorphisms in particular.

Thoroughly revised. Added some material: Continuous functional calculus, the Banach algebra of definable predicates, a little bit about definable sets, some examples of co-axiomatizable properties