KFU. Card publication. Differences and commutators of idempotents in C*-algebras

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FEDERAL UNIVERSITY

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DIFFERENCES AND COMMUTATORS OF IDEMPOTENTS IN C*-ALGEBRAS

Form of presentation

Articles in international journals and collections

Year of publication

2021

Язык

английский

Authors, employees of KFU

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. 13--22.

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 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.

Keywords

Hilbert space, linear operator, idempotent, tripotent, isoclinic projection, commutator, similarity, C*-algebra, trace, determinant

The name of the journal

RUSSIAN MATHEMATICS

On-line 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

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F_Bikchentaev_Fawwaz2021_Article_DifferencesAndCommutatorsOfIde.pdf	0,22	pdf	show / download

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	2021-01-01T00:00:00Z	ru_RU
dc.date.available	2021-01-01T00:00:00Z	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. 13--22.	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 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.	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