Free Convex Algebraic Geometry

Abstract

This chapter is a tutorial on techniques and results in free convex algebraic geometry and free real algebraic geometry (RAG). The term free refers to the central role played by algebras of noncommutative polynomials R<x> in free (freely noncommuting) variables x=(x_1,...,x_g). The subject pertains to problems where the unknowns are matrices or Hilbert space operators as arise in linear systems engineering and quantum information theory. The subject of free RAG flows in two branches. One, free positivity and inequalities is an analog of classical real algebraic geometry, a theory of polynomial inequalities embodied in algebraic formulas called Positivstellensätze; often free Positivstellensätze have cleaner statements than their commutative counterparts. Free convexity, the second branch of free RAG, arose in an effort to unify a torrent of ad hoc optimization techniques which came on the linear systems engineering scene in the mid 1990's. Mathematically, much as in the commutative case, free convexity is connected with free positivity through the second derivative: A free polynomial is convex if and only if its Hessian is positive. However, free convexity is a very restrictive condition, for example, free convex polynomials have degree 2 or less. This article describes for a beginner techniques involving free convexity. As such it also serves as a point of entry into the larger field of free real algebraic geometry.

70 pages, survey

References in corpus (10)

• A Matrix Convexity Approach to Some Celebrated Quantum Inequalities
• Connes' embedding conjecture and sums of hermitian squares
• Proper Analytic Free Maps
• Analytic mappings between noncommutative pencil balls
• The convex Positivstellensatz in a free algebra
• Positive polynomials in scalar and matrix variables, the spectral theorem and optimization
• Noncommutative ball maps
• Progress in noncommutative function theory
• A local-global principle for linear dependence of noncommutative polynomials
• Semidefinite programming in matrix unknowns which are dimension free

Cited by in corpus (8)

• Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball
• Joint measurability of quantum effects and the matrix diamond
• Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets
• A local-global principle for linear dependence of noncommutative polynomials
• Quadratic Modules, C*-Algebras, and Free Convexity
• The Ramsey property for Operator spaces and noncommutative Choquet simplices
• Free Semialgebraic Geometry
• Abstract Operator Systems over the Cone of Positive Semidefinite Matrices