Spectrum of a C*-algebra
In mathematics, the spectrum of a C*-algebra or dual of a C*-algebra A, denoted Â, is the set of unitary equivalence classes of irreducible *-representations of A.One of the most important applications of this concept is to provide a notion of dual object for any locally compact group.This dual object is suitable for formulating a Fourier transform and a Plancherel theorem for unimodular separable locally compact groups of type I and a decomposition theorem for arbitrary representations of separable locally compact groups of type I.Then there is a natural homeomorphism This mapping is defined by I(x) is a closed maximal ideal in C(X) so is in fact primitive.It is known A is isomorphic to a finite direct sum of full matrix algebras: where min(A) are the minimal central projections of A.In fact, the topology on  is intimately connected with the concept of weak containment of representations as is shown by the following: The second condition means exactly that π is weakly contained in S. The GNS construction is a recipe for associating states of a C*-algebra A to representations of A.This conjecture was proved by James Glimm for separable C*-algebras in the 1961 paper listed in the references below.A C*-algebra A is of type I if and only if any separable factor representation of A is a finite or countable multiple of an irreducible one.
C*-algebraunitary equivalenceirreducible*-representationHilbert spacedimensionalspacestopological spacespectrum of a ringlocally compact groupFourier transformPlancherel theoremunimodularseparableTannaka–Krein dualitycompact topological groupsPontryagin dualitytopologyprimitive idealsidempotentKuratowski closure axiomssurjectiveZariski topologystatessingletonsGelfand dualcompactHausdorff spacenaturalhomeomorphismdiscrete topologylocally compacttopological groupsGNS constructionBorel spaceG. Mackeycomplete separable metric spaceJames GlimmconnectednilpotentLie groupssemi-simpleHeisenberg groupsPrimitive spectrumannihilatorsimple moduleif and only ifgroup C*-algebraJ. M. G. FellFunctional analysistopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological vectorBarrelledCompleteLocally convexReflexiveHahn–BanachRiesz representationClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluAdjointBoundedHilbert–SchmidtNormalTrace classTransposeUnboundedUnitaryBanach algebraOperator algebraGroup algebra of a locally compact groupVon Neumann algebraInvariant subspace problemMahler's conjectureHardy spaceSpectral theory of ordinary differential equationsHeat kernelIndex theoremCalculus of variationsFunctional calculusIntegral linear operatorJones polynomialTopological quantum field theoryNoncommutative geometryRiemann hypothesisDistributionGeneralized functionsApproximation propertyBalanced setChoquet theoryWeak topologyBanach–Mazur distanceTomita–Takesaki theorySpectral theory*-algebrasInvolution/*-algebraB*-algebraNoncommutative topologyProjection-valued measureSpectrumSpectral radiusOperator spaceGelfand–Mazur theoremGelfand–Naimark theoremGelfand representationPolar decompositionSingular value decompositionSpectral theoremSpectral theory of normal C*-algebrasIsospectraloperatorHermitian/Self-adjointKrein–Rutman theorem