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Gelfand-Naimark-Segal construction : ウィキペディア英語版
Gelfand–Naimark–Segal construction
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 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.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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