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Concept
Gelfand–Naimark–Segal construction
Summary
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.
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 multiplicative unit element this condition is equivalent to f(1) = 1.
For a representation π of a C-algebra A on a Hilbert space H, an element ξ is called a cyclic vector if the set of vectors
is norm dense in H, in which case π is called a cyclic representation. Any non-zero vector of an irreducible representation is cyclic. However, non-zero vectors in a general cyclic representation may fail to be cyclic.
Let π be a -representation of a C-algebra A on the Hilbert space H and ξ be a unit norm cyclic vector for π. Then
is a state of A.
Conversely, every state of A may be viewed as a vector state as above, under a suitable canonical representation.
The method used to produce a -representation from a state of A in the proof of the above theorem is called the GNS construction.
For a state of a C-algebra A, the corresponding GNS representation is essentially uniquely determined by the condition, as seen in the theorem below.
The GNS construction is at the heart of the proof of the Gelfand–Naimark theorem characterizing C*-algebras as algebras of operators.
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