Operator systems and convex sets with many normal cones

• DOI: 10.48550/arxiv.1606.03792
• Publication Date: 2016-01-01
• Authors: Weis, Stephan
• Keywords: Ground state; 52A20, 52B05, 47L07, 51D25, 47A12 (Primary), 81P16, 14P10 (Secondary); FOS: Physical sciences; 01 natural sciences; Exposed face; Coatom; Mathematics - Algebraic Geometry; Exposed ray; Mathematics - Metric Geometry; FOS: Mathematics; 0101 mathematics; Operator Algebras (math.OA); Algebraic Geometry (math.AG); Mathematical Physics; Operator system; Mathematics - Operator Algebras; Metric Geometry (math.MG); Mathematical Physics (math-ph); Convex support; Coorbitope; Mathématiques; Joint numerical range; Normal cone; Analyse mathématique; State space

The state space of an operator system of $n$-by-$n$ matrices has, in a sense, many normal cones. Merely this convex geometrical property implies smoothness qualities and a clustering property of exposed faces. The latter holds since each exposed face is an intersection of maximal exposed faces. An isomorphism translates these results to the lattice of ground state projections of the operator system. We work on minimizing the assumptions under which a convex set has the mentioned properties.<br/>20 pages, one figure, discrepancy to v1: minor corrections, discussion of operator systems added (-> change of title)<br/>