Positive linear functional

In mathematics, more specifically in functional analysis, a positive linear functional on an ordered vector spaceIn other words, a positive linear functional is guaranteed to take nonnegative values for positive elements.The significance of positive linear functionals lies in results such as Riesz–Markov–Kakutani representation theorem.is a complex vector space, it is assumed that for allis a C*-algebra with its partially ordered subspace of self-adjoint elements, sometimes a partial order is placed on only a subspaceand the partial order does not extend to all ofin which case the positive elements ofThis implies that for a C*-algebra, a positive linear functional sends anyto a real number, which is equal to its complex conjugate, and therefore all positive linear functionals preserve the self-adjointness of suchThis property is exploited in the GNS construction to relate positive linear functionals on a C*-algebra to inner products.There is a comparatively large class of ordered topological vector spaces on which every positive linear form is necessarily continuous.[1] This includes all topological vector lattices that are sequentially complete.be an Ordered topological vector space with positive conedenote the family of all bounded subsets ofThen each of the following conditions is sufficient to guarantee that every positive linear functional onis continuous: The following theorem is due to H. Bauer and independently, to Namioka.[1] Proof: It suffices to endowwith the finest locally convex topology makingThe trace function defined on this C*-algebra is a positive functional, as the eigenvalues of any positive-definite matrix are positive, and so its trace is positive.of all continuous complex-valued functions of compact support on a locally compact Hausdorff spaceConsider a Borel regular measureMoreover, any positive functional on this space has this form, as follows from the Riesz–Markov–Kakutani representation theorem.be a C*-algebra (more generally, an operator system in a C*-algebradenote the set of positive elements inis a positive linear functional on a C*-algebrathen one may define a semidefinite sesquilinear form on, a price system can be viewed as a continuous, positive, linear functional on
mathematicsfunctional analysisordered vector spacelinear functionalpositive elementsRiesz–Markov–Kakutani representation theoremcomplexC*-algebraGNS constructioninner productsordered topological vector spacestopological vector latticessequentially completeOrdered topological vector spacepositive conecompletemetrizablebornologicalsemi-completestrict B {\displaystyle {\mathcal {B}}} -coneinductive limitFréchet spacesabsorbing subsetsquare matricespositive-definite matriceseigenvaluesRiesz spacecontinuouscompactsupportlocally compactHausdorff spaceBorel regular measureoperator systemsesquilinear formCauchy–Schwarz inequalityPositive elementPositive linear operatorKadison, RichardSchaefer, Helmut H.Trèves, FrançoistopicsglossaryBanachFréchetHilbertHölderNuclearOrliczSchwartzSobolevTopological vectorBarrelledLocally convexReflexiveSeparableHahn–BanachRiesz representationClosed graphUniform boundedness principleKrein–MilmanMin–maxGelfand–NaimarkBanach–AlaogluAdjointBoundedHilbert–SchmidtNormalTrace classTransposeUnboundedUnitaryBanach algebraSpectrum of a C*-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 theoryPartially ordered spaceOrder topologyOrder unitTopological vector latticeVector latticeAL-spaceAM-spaceArchimedeanBanach latticeFréchet latticeLocally convex vector latticeNormed latticeOrder bound dualOrder dualOrder completeRegularly orderedCone-saturatedLattice disjointDual/Polar coneNormal coneOrder summableQuasi-interior pointSolid setWeak order unitOrder convergenceFreudenthal spectral