positive linear functional

0.0.1 Definition

Let $𝒜$ be a $C^{*}$-algebra (http://planetmath.org/CAlgebra) and $\varphi$ a linear functional on $𝒜$.

We say that $\varphi$ is a positive linear functional on $𝒜$ if $\varphi$ is such that $\varphi (x)\ge 0$ for every $x\ge 0$, i.e. for every positive element $x\in 𝒜$.

0.0.2 Properties

Let $\varphi$ be a positive linear functional on $𝒜$. Then

• $\varphi (x^{*})=\overline{\varphi (x)}$ for every $x\in 𝒜$.

• ${|\varphi (x^{*}y)|}^2\le \varphi (x^{*}x)\varphi (y^{*}y)$ for every $x,y\in 𝒜$. This is an analog of the Cauchy-Schwartz inequality

Let $\varphi$ be a linear functional on a $C^{*}$-algebra $𝒜$ with identity element $e$. Then

• $\varphi$ is positive if and only if $\varphi$ is bounded (http://planetmath.org/ContinuousLinearMapping) and $\parallel \varphi \parallel =\varphi (e)$.

0.0.3 Examples

• Let $X$ be a locally compact Hausdorff space and $C_0(X)$ the $C^{*}$-algebra of continuous functions $X⟶ℂ$ that vanish at infinity. Let $\mu$ be a regular Radon measure on $X$. The linear functional $\varphi$ defined by integration against $\mu$,

  $$\varphi (f):={\int }_{X}f𝑑\mu ,f\in C_0(x)$$

  is a positive linear functional on $C_0(X)$. In fact, by the Riesz representation theorem (http://planetmath.org/RieszRepresentationTheoremOfLinearFunctionalsOnFunctionSpaces), all positive linear functionals of $C_0(X)$ are of this form.