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Статья
2021

Differences and Commutators of Idempotents in \(C^*\)-Algebras


A. M. BikchentaevA. M. Bikchentaev, Kh. FawwazKh. Fawwaz
Русская математика
https://doi.org/10.3103/S1066369X21080028
Abstract / Full Text

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.

Author information
  • Kazan Federal University, 18 Kremlyovskaya str., 420008, Kazan, RussiaA. M. Bikchentaev & Kh. Fawwaz
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