Differences and commutators of idempotents in $C^*$-algebras

We establish similarity between some tripotents and idempotents on a Hilbert space $\mathcal{H}$ and obtain new results on differences and commutators of idempotents $ P $ and $ Q $. In the unital case, the difference $ P-Q $ is associated with the difference $A_{P, Q}$ of another pair of idempotents. Let $\varphi $ be a trace on a unital $C^*$-algebra $\mathcal{A}$, $\mathfrak{M}_{\varphi} $ be the ideal of definition of the trace $\varphi $. If $ P-Q \in \mathfrak{M}_\varphi $, then $ A_{P, Q} \in \mathfrak {M}_\varphi $ and $ \varphi (A_{P, Q}) = \varphi (P-Q) \in \mathbb{R}$. In some cases, this allowed us to establish the equality $ \varphi (P-Q) = 0$. We obtain new identities for pairs of idempotents and for pairs of isoclinic projections. It is proved that each operator $ A \in \mathcal{B} (\mathcal{H}) $, $ \dim \mathcal{H} = \infty $, can be presented as a sum of no more than 50 commutators of idempotents from $ \mathcal{B} (\mathcal{H}) $. It is shown that the commutator of an idempotent and an arbitrary element from an algebra $ \mathcal{A} $ cannot be a nonzero idempotent. If $ \mathcal{H} $ is separable and $ \dim \mathcal{H} = \infty $, then each skew-Hermitian operator $ T \in \mathcal {B} (\mathcal{H}) $ can be represented as a sum $ T = \sum_{k = 1}^4 [A_k, B_k] $, where $ A_k, B_k \in \mathcal{B} (\mathcal {H}) $ are skew-Hermitian.