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Journal of Convex Analysis 25 (2018), No. 1, 041--064
Copyright Heldermann Verlag 2018

Operator Systems and Convex Sets with Many Normal Cones

Stephan Weis
Departamento de Matemática, Inst. Matemática e Estatística, Univ. Estadual de Campinas, Rua S. Buarque de Holanda 651, Campinas-SP / CEP 13083-859, Brazil
and: Centre for Quantum Information, Ecole Polytechnique, Université Libre, 50 av. F.D. Roosevelt - CP165/59, 1050 Bruxelles, Belgium

The state space of an operator system of n-by-n matrices has, in a sense, many normal cones. This wealth of normal cones is a smoothness quality and it implies that each exposed face is an intersection of maximal exposed faces. A clustering property of exposed faces follows. We work on minimizing the assumptions which allow an arbitrary convex set of finite dimension to have analogous properties. One motivation for this work is an isomorphism from the exposed faces of the state space to the ground state spaces of the operator system.

Keywords: Normal cone, exposed ray, exposed face, coatom, operator system, state space, convex support, joint numerical range, ground state, coorbitope.

MSC: 52A20, 52B05, 47L07, 51D25, 47A12; 81P16, 14P10

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