Foundational aspects of uncountable measure theory: gelfand duality, riesz representation, canonical models, and canonical disintegration
Publication Date
2023
Advisor
Institution Author
Jamneshan, Asgar
Co-Authors
Tao, Terence
Journal Title
Journal ISSN
Volume Title
Publisher:
Institute of Mathematics. Polish Academy of Sciences
Type
Journal Article
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 present a canonical model that represents 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 are useful in applications to “uncountable” ergodic theory (as demonstrated by the authors and others). © 2023 Institute of Mathematics. Polish Academy of Sciences. All rights reserved.
Description
Subject
Mathematics