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Differences and commutators of idempotents in C*-algebras

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dc.contributor Казанский федеральный университет
dc.contributor.author Bikchentaev Airat Midkhatovich
dc.contributor.author Fawwaz Khattab Omar
dc.date.accessioned 2021-09-06T08:49:24Z
dc.date.available 2021-09-06T08:49:24Z
dc.date.issued 2021
dc.identifier.citation Bikchentaev A.M., Fawwaz Kh. Differences and commutators of idempotents in C*-algebras / A.M. Bikchentaev, Kh. Fawwaz // Russian Mathematics. - 2021. - Vol. 65. - No. 8. - P. 13--22.
dc.identifier.uri https://dspace.kpfu.ru/xmlui/handle/net/165844
dc.description.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.
dc.language.iso en
dc.relation.ispartofseries RUSSIAN MATHEMATICS
dc.rights открытый доступ
dc.subject Hilbert space
dc.subject linear operator
dc.subject idempotent
dc.subject tripotent
dc.subject isoclinic projection
dc.subject commutator
dc.subject similarity
dc.subject C*-algebra
dc.subject trace
dc.subject determinant
dc.subject.other Математика
dc.title Differences and commutators of idempotents in C*-algebras
dc.type Article
dc.contributor.org Институт математики и механики им. Н.И. Лобачевского
dc.description.pages 13-22
dc.relation.ispartofseries-issue 8
dc.relation.ispartofseries-volume 65
dc.pub-id 255872
dc.identifier.doi 10.3103/S1066369X21080028

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