Finite-dimensional C*-algebras - 02 - positivity
Channel: Arthur Parzygnat
Date: 20/04/2020

Positive maps and states on finite-dimensional C*-algebras are introduced. The main examples are on function spaces over finite sets and matrix algebras. States on matrix algebras correspond to density matrices. 

Notes: 
1. To clarify, when I say "positive," I mean "positive semi-definite." The terminology "positive" is used more often in the literature on C*-algebras.  
2. In the first example, I proved that f=g*g implies f(x)\ge0 for all x. I didn't prove the converse, but this follows from defining g(x) to be the square-root of f(x). 
3. In the second example for matrix algebras, one needs to assume that A is self-adjoint in the second "iff," so it should read "iff A is self-adjoint and (v,Av)\ge0 for all v."
4. I said that I would explain why the last 1-1 correspondence holds "shortly", but I was unfortunately unable to make a video for that. One proof uses the Hilbert--Schmidt (aka Frobenius) inner product on matrix algebras. This is what relates PU (positive unital) maps (Heisenberg picture) to PTP (positive trace-preserving) maps (Schrodinger picture). Once one proves this, it follows that a state corresponds to a PTP map from C to your matrix algebra. But a linear map from C to any algebra picks a single matrix (the image of 1). By the positivity condition, this matrix is positive. By the trace-preserving condition, this matrix has trace 1. That's the density matrix. See Remark 3.15 in https://arxiv.org/abs/1907.09689v1 for details (this is also discussed in many textbooks). 

Reference: Attal's book "Lectures in Quantum Noise Theory" http://math.univ-lyon1.fr/~attal/chapters.html, specifically Lecture 6, reviews many of these things in detail and with greater generality. 
Another 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 next video on Special Topics in Linear Algebra is on the Stochastic Gelfand--Naimark theorem: https://youtu.be/EdxGXrdZ7ws

The previous video on Special Topics in Linear Algebra is *-homomorphisms: https://youtu.be/NbgpY0hkFi0

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