positive linear functional

0.0.1 Definition

Let 𝒜 be a C*-algebra (http://planetmath.org/CAlgebra) and ϕ a linear functional on 𝒜.

We say that ϕ is a positive linear functionalMathworldPlanetmath on 𝒜 if ϕ is such that ϕ(x)0 for every x0, i.e. for every positive elementMathworldPlanetmathPlanetmathPlanetmath x𝒜.

0.0.2 Properties

Let ϕ be a positive linear functional on 𝒜. Then

  • ϕ(x*)=ϕ(x)¯ for every x𝒜.

  • |ϕ(x*y)|2ϕ(x*x)ϕ(y*y) for every x,y𝒜. This is an analog of the Cauchy-Schwartz inequality

Let ϕ be a linear functional on a C*-algebra 𝒜 with identity elementMathworldPlanetmath e. Then

  • ϕ is positive if and only if ϕ is boundedPlanetmathPlanetmathPlanetmathPlanetmath (http://planetmath.org/ContinuousLinearMapping) and ϕ=ϕ(e).

0.0.3 Examples

Title positive linear functional
Canonical name PositiveLinearFunctional
Date of creation 2013-03-22 17:45:05
Last modified on 2013-03-22 17:45:05
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 11
Author asteroid (17536)
Entry type Definition
Classification msc 46L05