*-representation

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

A state on 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 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.

In fact, 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, ${\displaystyle \rho (a)=\langle \pi (a)\xi ,\xi \rangle }$ as seen in the theorem below.

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