Finite-dimensional C*-algebras - 03 - Stochastic Gelfand--Naimark theorem
Channel: Arthur Parzygnat
Date: 2020-04-20

The Stochastic Gelfand--Naimark theorem says that stochastic maps (conditional probabilities) between spaces transfer to positive maps on the corresponding function algebras. It actually says more than what I was able to discuss in this video here. It would be helpful if the viewer is familiar with my videos on probability theory: https://www.youtube.com/playlist?list=PLSx1kJDjrLRQksb7H9fqRE8GVMJdkX-4A

Note: I didn't prove that the composition of stochastic maps goes to composition of positive maps because I wanted to focus on the full and faithful properties of the functor going from spaces to algebras. 

Reference: Chapter 2 of https://arxiv.org/abs/1708.00091

This is part of a series of lectures on special topics in linear algebra https://www.youtube.com/playlist?list=PLSx1kJDjrLRSS4Aio9WfQlNnH6X6pC_8t. It is assumed the viewer has taken (or is well into) a course in linear algebra. Some topics may also require additional background. Topics covered include linear regression and data analysis, ordinary linear differential equations, differential operators, function spaces, category-theoretic aspects of linear algebra, support vector machines (machine learning), Hamming's error-correcting codes, stochastic maps and Markov chains, tensor products, finite-dimensional C*-algebras, algebraic probability theory, completely positive maps, aspects of quantum information theory, and more.

The previous video on Special Topics in Linear Algebra is on positivity: https://youtu.be/5dkEkq4LF_g

Stochastic maps are discussed in https://youtu.be/xhDCmyDBj24

These videos were created during the 2019 Spring/Summer semester at the UConn CETL Lightboard Room.