Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration

Abstract

We collect several foundational results regarding the interaction between locally compact spaces, probability spaces and probability algebras, and commutative $C^*$-algebras and von Neumann algebras equipped with traces, in the "uncountable" setting in which no separability, metrizability, or standard Borel hypotheses are placed on these spaces and algebras. In particular, we review the Gelfand dualities and Riesz representation theorems available in this setting. We also introduce a canonical model that represents (opposite) probability algebras as compact Hausdorff probability spaces in a completely functorial fashion, and apply this model to obtain a canonical disintegration theorem and to readily construct various product measures. These tools will be used in future papers by the authors and others in various applications to "uncountable" ergodic theory.