C*-algebras | Functional analysis

State (functional analysis)

In functional analysis, a state of an operator system is a positive linear functional of norm 1. States in functional analysis generalize the notion of density matrices in quantum mechanics, which represent quantum states, both §§ Mixed states​ and Pure states. Density matrices in turn generalize state vectors, which only represent pure states. For M an operator system in a C*-algebra A with identity, the set of all states of M, sometimes denoted by S(M), is convex, weak-* closed in the Banach dual space M*. Thus the set of all states of M with the weak-* topology forms a compact Hausdorff space, known as the state space of M . In the C*-algebraic formulation of quantum mechanics, states in this previous sense correspond to physical states, i.e. mappings from physical observables (self-adjoint elements of the C*-algebra) to their expected measurement outcome (real number). (Wikipedia).

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State (functional analysis) | Choquet theory | Density matrix | Operator system | Hahn decomposition theorem | Positive linear functional | Signed measure | Gelfand representation | Functional analysis | Operator norm | Radon measure | Probability measure | Net (mathematics) | Von Neumann algebra | C*-algebra | Gelfand–Naimark–Segal construction