Differences and Commutators of Idempotents in C* -Algebras

Abstract

We establish similarity between some tripotents and idempotents on a Hilbert space $\mathcal{H}$ andobtain 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 $\mathcal{A}$, $\mathfrak{M}_{\varphi}$ bethe 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 50commutators 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 anonzero 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 asum $T = \sum_{k = 1}^4 [A_k, B_k]$, where $A_k, B_k \in \mathcal{B} (\mathcal {H})$ are skew-Hermitian.