Gelfand–Naimark–Segal construction

In functional analysis, a discipline within mathematics, given aIt is named for Israel Gelfand, Mark Naimark, and Irving Segal.has a multiplicative unit element this condition is equivalent to, in which case π is called a cyclic representation.Any non-zero vector of an irreducible representation is cyclic.may be viewed as a vector state as above, under a suitable canonical representation.Since positive linear functionals are bounded, the equivalence classes of the netin the proof of the above theorem is called the GNS construction., the corresponding GNS representation is essentially uniquely determined by the condition,that implements the unitary equivalence mapsThe GNS construction is at the heart of the proof of the Gelfand–Naimark theorem characterizing-algebra has sufficiently many pure states (see below) so that the direct sum of corresponding irreducible GNS representations is faithful.-representation is a direct sum of cyclic representations, it follows that everyis a direct summand of some sum of copies of the universal representation.in the weak operator topology is called the enveloping von Neumann algebra of-representations and extreme points of the convex set of states.with a unit element is a compact convex set under the weak-has a unit element) the set of positive functionals of normBoth of these results follow immediately from the Banach–Alaoglu theorem., Riesz–Markov–Kakutani representation theorem says that the positive functionals of normIt follows from Krein–Milman theorem that the extremal states are the Dirac point-mass measures.is an extreme point of the convex set of positive linear functionals onTo prove this result one notes first that a representation is irreducible if and only if the commutant of, consists of scalar multiples of the identity.as a sum of positive linear functionals:The Stinespring factorization theorem characterizing completely positive maps is an important generalization of the GNS construction.Gelfand and Naimark's paper on the Gelfand–Naimark theorem was published in 1943.[3] Segal recognized the construction that was implicit in this work and presented it in sharpened form.[4] In his paper of 1947 Segal showed that it is sufficient, for any physical system that can be described by an algebra of operators on a Hilbert space, to consider the irreducible representations of aThis, as Segal pointed out, had been shown earlier by John von Neumann only for the specific case of the non-relativistic Schrödinger-Heisenberg theory.
functional analysismathematics C ∗ {\displaystyle C^{*}} -algebralinear functionalsIsrael GelfandMark NaimarkIrving SegalHilbert spacemappingbounded operatorsring homomorphisminvolutionnondegeneratepositive linear functionalirreducible representationsesquilinear formleft idealquotient spaceCauchy completionadjoint of an operatorapproximate identityGelfand–Naimark theoremfaithfuluniversal representationweak operator topologyenveloping von Neumann algebrairreducibleconvex setBanach–Alaoglu theoremRiesz–Markov–Kakutani representation theoremKrein–Milman theoremextreme pointcommutantRadon–Nikodym theorem B ∗ {\displaystyle B^{*}} -algebrasStinespring factorization theoremcompletely positive mapsJohn von NeumannCyclic and separating vectorWilliam ArvesonKadison, RichardFundamentals of the Theory of Operator AlgebrasJacques Dixmierdeformation quantizationG. SardanashvilyShoichiro SakaiKadison, R. V.I. M. GelfandM. A. NaimarkMatematicheskii SbornikRichard V. KadisonI. E. Segal