Differences and commutators of idempotents in C*-algebras

Abstract

We establish similarity between some tripotents and idempotents on a Hilbert space $H$ 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 $A$, $M_{\varphi}$ be the ideal of definition of the trace $\varphi$. If $P - Q\in M_\varphi$, then $A_{P,Q} \in M_\varphi$ 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 $A \in B (H)$, $\dim H = \infty$, can be presented as a sum of no more than 50 commutators of idempotents from $B(H)$. It is shown that the commutator of an idempotent and an arbitrary element from an algebra $A$ cannot be a nonzero idempotent. If $H$ is separable and $\dim H = \infty$, then each skew-Hermitian operator $T \in B (H)$ can be represented as a sum $T = \sum_{k=1}^4 [A_k, B_k]$, where $A_k, B_k \in B (H)$ are skew-Hermitian.