Trace and Differences of Idempotents in $C^*$-Algebras

Let $\varphi$ be a trace on a unital $C^*$-algebra $\mathcal{A}$, let $\mathfrak{M}_{\varphi}$ be the ideal of definition of the trace $\varphi$, and let $P,Q \in \mathcal{A}$ be idempotents such that $QP=P$. If $Q \in \mathfrak{M}_{\varphi}$, then $P \in \mathfrak{M}_{\varphi}$ and $0 \le \varphi(P) \le \varphi(Q)$. If $Q-P \in \mathfrak{M}_{\varphi}$, then $\varphi(Q-P)\in \mathbb{R}^+$. Let $A,B\in \mathcal{A}$ be tripotents. If $AB=B$ and $A\in \mathfrak{M}_{\varphi}$, then $B \in \mathfrak{M}_{\varphi}$ and $0 \le \varphi (B^2)\le \varphi (A^2)<+\infty$. Let $\mathcal{A}$ be a von Neumann algebra. Then
$$ \varphi(|PQ-QP|)\le \min\{\varphi(P),\varphi(Q),\varphi(|P-Q|)\} $$
for all projections $P,Q \in \mathcal{A}$. The following conditions are equivalent for a positive normal functional $\varphi$ on a von Neumann algebra $\mathcal{A}$:
(i) $\varphi $ is a trace;
(ii) $\varphi(Q-P) \in \mathbb{R}^+$ for all idempotents $P,Q \in \mathcal{A}$ with $QP=P$;
(iii) $ \varphi(|PQ-QP|) \le \min\{\varphi(P),\varphi(Q)\}$ for all projections $P,Q \in \mathcal{A}$;
(iv) $\varphi(PQ+QP) \le \varphi(PQP+QPQ)$ for all projections $P,Q \in \mathcal{A}$.