Let $M\subseteq B(H)$ be a von Neumann algebra. Let $\omega_1, \omega_2$ be functionals on $M$ satisfying $$\omega_1(p) \le \omega_2(p) \quad (*)$$ for all projections $p \in M$. Is it true that $\omega_1 \le \omega_2$, i.e. that $(*)$ holds for all positive elements $p \in M$ as well?
I think I need some decomposition result that says that a positive elements is a linear combination with positive coefficients of projections. Do we have such result?
Let $x\in M$ be positive and $e$ its spectral measure (the projection-valued version). By the spectral theorem, $$ x=\int_{[0\|x\|]}\lambda\,de(\lambda). $$ For every $\epsilon>0$ there exists a tagged partition $(I_k,\lambda_k)_{1\leq k\leq n}$ of $[0,\|x\|]$ such that $|\lambda-\lambda_k|<\epsilon$ for $\lambda\in I_k$. Then $y=\sum_{k=1}^n \lambda_k 1_{I_k}(x)$ is a positive linear combination of projections in $M$ and $$ \|x-y\|=\left\lVert\sum_{k=1}^n\int_{[0,\|x\|]}(\lambda-\lambda_k)\,d e(\lambda)\right\rVert\leq \epsilon. $$ In particular, $x$ is the norm limit of positive linear combinations of projections.