Concept

Gelfand–Naimark–Segal construction

Résumé
In functional analysis, a discipline within mathematics, given a C*-algebra A, the Gelfand–Naimark–Segal construction establishes a correspondence between cyclic *-representations of A and certain linear functionals on A (called states). The correspondence is shown by an explicit construction of the *-representation from the state. It is named for Israel Gelfand, Mark Naimark, and Irving Segal. States and representations A -representation of a C-algebra A on a Hilbert space H is a mapping π from A into the algebra of bounded operators on H such that
  • π is a ring homomorphism which carries involution on A into involution on operators
  • π is nondegenerate, that is the space of vectors π(x) ξ is dense as x ranges through A and ξ ranges through H. Note that if A has an identity, nondegeneracy means exactly π is unit-preserving, i.e. π maps the identity of A to the identity operator on H.
A state on a C*-algebra A is a positive linear functional f of norm 1. If A has a
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