An introduction to matrix convex sets and free spectrahedra

DOI10.1007/s11785-019-00937-8zbMath1435.14052arXiv1611.03103OpenAlexW2795723564WikidataQ127539731 ScholiaQ127539731MaRDI QIDQ2008156

Tom-Lukas Kriel

Publication date: 22 November 2019

Published in: Complex Analysis and Operator Theory (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1611.03103

zbMATH Keywords

Gleichstellensatzmatrix convexitymatrix extreme pointfree spectrahedraabsolute extreme point

Mathematics Subject Classification ID

Axiomatic and generalized convexity (52A01) Operator spaces and completely bounded maps (46L07) Semialgebraic sets and related spaces (14P10) Convex sets and cones of operators (47L07) Real algebra (13J30)

Related Items (11)

Passive linear continuous-time systems: characterization through structure ⋮ Facial structure of matrix convex sets ⋮ Complex free spectrahedra, absolute extreme points, and dilations ⋮ Noncommutative partial convexity via \(\Gamma\)-convexity ⋮ Empirical Properties of Optima in Free Semidefinite Programs ⋮ Compressions of compact tuples ⋮ Complete order equivalence of spin unitaries ⋮ Noncommutative polynomials describing convex sets ⋮ Bianalytic free maps between spectrahedra and spectraballs ⋮ Dilation theory in finite dimensions and matrix convexity ⋮ The Arveson boundary of a free quadrilateral is given by a noncommutative variety

Cites Work

• Unnamed Item
• Unnamed Item
• Unnamed Item
• Unnamed Item
• Operator Positivstellensätze for noncommutative polynomials positive on matrix convex sets
• The convex Positivstellensatz in a free algebra
• Every convex free basic semi-algebraic set has an LMI representation
• Free loci of matrix pencils and domains of noncommutative rational functions
• Obstructions to determinantal representability
• Matrix convexity: Operator analogues of the bipolar and Hahn-Banach theorems
• Semidefinite representation of convex sets
• The Choquet boundary of an operator system
• Geometry of free loci and factorization of noncommutative polynomials
• Extreme points of matrix convex sets, free spectrahedra, and dilation theory
• Matrix convex sets without absolute extreme points
• Completely rank-nonincreasing linear maps
• Some geometric results in semidefinite programming
• The matricial relaxation of a linear matrix inequality
• On the exactness of Lasserre relaxations and pure states over real closed fields
• Arveson extreme points span free spectrahedra
• The tracial Hahn-Banach theorem, polar duals, matrix convex sets, and projections of free spectrahedra
• Matrix convex hulls of free semialgebraic sets
• Chapter 8: Free Convex Algebraic Geometry
• The noncommutative Choquet boundary
• Generators of the Ring of Bounded Operators
• Linear matrix inequality representation of sets
• Sufficient and Necessary Conditions for Semidefinite Representability of Convex Hulls and Sets
• Non-commutative partial matrix convexity
• A Proof of the Boundary Theorem
• Nonseparable Approximate Equivalence
• The Krein-Milman theorem in operator convexity
• Invariant Subspaces for Pairs of Projections
• The Structure of C^*-Convex Sets
• Linear Matrix Inequalities in System and Control Theory
• Dilations, Inclusions of Matrix Convex Sets, and Completely Positive Maps
• Convex Matrix Inequalities Versus Linear Matrix Inequalities
• Spectrahedral Containment and Operator Systems with Finite-Dimensional Realization
• Dilations, Linear Matrix Inequalities, the Matrix Cube Problem and Beta Distributions
• \(C^*\)-algebras by example

This page was built for publication: An introduction to matrix convex sets and free spectrahedra