Form of presentation  Articles in international journals and collections 
Year of publication  2021 
Язык  английский 

Bikchentaev Ayrat Midkhatovich, author
Fauaz Khattab , author

Bibliographic description in the original language 
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. 1322.

Annotation 
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 skewHermitian
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 skewHermitian. 
Keywords 
Hilbert space, linear operator, idempotent, tripotent, isoclinic projection,
commutator, similarity, C*algebra, trace, determinant 
The name of the journal 
RUSSIAN MATHEMATICS

Online resource for training course 
http://dspace.kpfu.ru/xmlui/bitstream/handle/net/165844/F_Bikchentaev_Fawwaz2021_Article_DifferencesAndCommutatorsOfIde.pdf?sequence=1&isAllowed=y

Please use this ID to quote from or refer to the card 
https://repository.kpfu.ru/eng/?p_id=255872&p_lang=2 
Resource files  

Full metadata record 
Field DC 
Value 
Language 
dc.contributor.author 
Bikchentaev Ayrat Midkhatovich 
ru_RU 
dc.contributor.author 
Fauaz Khattab 
ru_RU 
dc.date.accessioned 
20210101T00:00:00Z 
ru_RU 
dc.date.available 
20210101T00:00:00Z 
ru_RU 
dc.date.issued 
2021 
ru_RU 
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. 1322.

ru_RU 
dc.identifier.uri 
https://repository.kpfu.ru/eng/?p_id=255872&p_lang=2 
ru_RU 
dc.description.abstract 
RUSSIAN MATHEMATICS 
ru_RU 
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 skewHermitian
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 skewHermitian. 
ru_RU 
dc.language.iso 
ru 
ru_RU 
dc.subject 
Hilbert space 
ru_RU 
dc.subject 
linear operator 
ru_RU 
dc.subject 
idempotent 
ru_RU 
dc.subject 
tripotent 
ru_RU 
dc.subject 
isoclinic projection 
ru_RU 
dc.subject 
commutator 
ru_RU 
dc.subject 
similarity 
ru_RU 
dc.subject 
C*algebra 
ru_RU 
dc.subject 
trace 
ru_RU 
dc.subject 
determinant 
ru_RU 
dc.title 
Differences and commutators of idempotents in C*algebras 
ru_RU 
dc.type 
Articles in international journals and collections 
ru_RU 
