Let $M$ be a von Neumann algebra. Suppose that $\rho$ is a normal linear functional on $M$. We have the Jordan decomposition :$\rho=\omega_{+}-\omega_{-}$ and $\|\rho\|=\|\omega_{+}\|+\|\omega_{-}\|$, where $\omega_{+}$ and $\omega_{-}$ are positive normal functionals on $M$ with orthogonal supports $e$ and $f$.
My question: 1.Can we express $\omega_{+}$ in terms of $\rho$?
2.Is there any relationship between the support of $\omega_{+}$ and the support of $\rho$?
- I think the above decomposition is not true for zero linear functional.
$\rho$ has to be Hermitian in addition to normal. I'm not sure what you mean by 'express', but I think $e$ is the supremum of all projections $E$ in $M$ such that $x\mapsto \rho(ExE)$ is positive definite. Then $\rho_+ (x) = \rho(e x e)$. Similary $f$ is the supremum of all projections $F$ in $M$ such that $x \mapsto -\rho(FxF)$ is positive definite and $\rho_-(x) = -\rho(fxf)$. The support of $\rho$ is $e+f$. For your last question if $\rho = 0$ then $e=f=0$ and $\rho_+ = \rho_- = 0$. It is not required that $e+f = I$.
Edit: by positive definite I mean that $ExE \ge 0 \implies \rho(ExE)\ge 0$ and if additionally $ExE \ne 0$ then $\rho(ExE) > 0$.