Let $\Omega$ be a locally compact and Hausdorff topological space. The Riesz representation theorem says that $C\_0(\Omega)^\*$ , dual of the commutative C\*-algebra $C\_0(\Omega)$, is just the space of complex Radon measures $M(\Omega)$.


$$\gamma: M(\Omega)\simeq C\_0(\Omega)^\* : \gamma(\mu)(f)=\int f d\mu$$ 


Let $\mu$ be a complex Radon measure and denote $[\mu]$, by the total variation of $\mu$. 


Question: It seems that $\gamma([\mu])=|\gamma(\mu)|$, where $|\gamma(\mu|)$ is the absolute value of the bounded linear functional $\gamma(\mu)$ on $C\_0(\Omega)$ ?



The total variation of a complex measure
The answer is directly obtained by the following interesting fact: 


Theorem. Let $M$ be a von neumann algebra and $f$ be a normal functional on $M$. 
Assume $a$ is in the unit ball of $M$ with $||af||=|f||$ and $af$ is positive. Then $af$ is just the absolute value of $|f|$. (see theorem 3.2 in Order ideals in a C\*-algebra and its dual by E. Effros 1962)


