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Operational Quantum Logic

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Introductory lecture summary:

Operational Quantum Logic I: Effect Algebras, States, and Basic Convexity
• Effect algebras, effect test-spaces, PAS's (partial abelian semigroups).
• Morphisms, states, dynamics. Classes of effect algebras whose state-set has nice properties.
• Operational derivation of effect algberas, summarized.
• "Theories"--- Effect-state systems.
• Tensor product (defined, existence result stated).
• Some notions of sharpness in EA's, examples that separate them, conditional equivalences that are interesting.
• Convex cones/sets, ordered linear space basics. Partially ordered abelian groups.
Operational Quantum Logic II: Convexity, Representations, and Operations
• Convex cones and convex sets. Extremality. Krein-Milman. Caratheodory. Affine maps.
• Positive maps. Automorphisms. Dual space, Dual cone. Adjoint map. Faces. Exposed faces. Lattices of faces.
• Interval EA's, representations on partially ordered abelian groups, unigroups. Analogues of Naimark's theorem, open problems.
• Convex EA's. Observables, "generalized" observables. Representation theorem for convex EA's. Relation of observables to effects formulation.
• State representation theorem for finite-d homogeneous self-dual cones (statement).
• Homogeneous cones as slices of positive semidefinite cones (statement).
• Axioms concerning the face lattice