Form of presentation  Articles in international journals and collections 
Year of publication  2023 
Язык  английский 

Bikchentaev Ayrat Midkhatovich, author

Bibliographic description in the original language 
Airat M. Bikchentaev, The algebra of thin measurable operators is directly finite // Constructive Mathematical Analysis. 2023.
V. 6, no 1. P. 15. 
Annotation 
Let $\mathcal{M}$ be a semifinite von Neumann algebra on a Hilbert space $\cH$ equipped with a faithful normal semifinite trace $\tau$, $S(\mathcal{M},\tau)$ be the ${}^*$algebra of all $\tau$measurable operators. Let $S_0(\mathcal{M},\tau)$ be the ${}^*$algebra of all $\tau$compact operators and
$T(\mathcal{M},\tau)=S_0(\mathcal{M},\tau)+\mathbb{C}I$ be the ${}^*$algebra of all operators $X=A+\lambda I$
with $A\in S_0(\mathcal{M},\tau)$ and $\lambda \in \mathbb{C}$. We prove that every operator of $T(\mathcal{M},\tau)$ that is leftinvertible in $T(\mathcal{M},\tau)$ is in fact invertible in $T(\mathcal{M},\tau)$.
It is a generalization of Sterling Berberian theorem (1982) on the subalgebra of thin operators in $\cB (\cH)$.
For the singular value function $\mu(t; Q)$ of $Q=Q^2\in S(\mathcal{M},\tau)$ we have $\mu(t; Q)\in \{0\}\bigcup
[1, +\infty)$ for all $t>0$. It gives the positive answer to the question posed by Daniyar Mushtari in 2010. 
Keywords 
Hilbert space, von Neumann algebra, semifinite trace, $\tau$measurable operator, $\tau$compact operator, singular value function, idempotent 
The name of the journal 
Constructive Mathematical Analysis

Online resource for training course 
http://dspace.kpfu.ru/xmlui/bitstream/handle/net/175023/10.33205_cma.1181495_2676924.pdf?sequence=1&isAllowed=y

URL 
http://dergipark.org.tr/en/pub/cma 
Please use this ID to quote from or refer to the card 
https://repository.kpfu.ru/eng/?p_id=276828&p_lang=2 
Resource files  

Full metadata record 
Field DC 
Value 
Language 
dc.contributor.author 
Bikchentaev Ayrat Midkhatovich 
ru_RU 
dc.date.accessioned 
20230101T00:00:00Z 
ru_RU 
dc.date.available 
20230101T00:00:00Z 
ru_RU 
dc.date.issued 
2023 
ru_RU 
dc.identifier.citation 
Airat M. Bikchentaev, The algebra of thin measurable operators is directly finite // Constructive Mathematical Analysis. 2023.
V. 6, no 1. P. 15. 
ru_RU 
dc.identifier.uri 
https://repository.kpfu.ru/eng/?p_id=276828&p_lang=2 
ru_RU 
dc.description.abstract 
Constructive Mathematical Analysis 
ru_RU 
dc.description.abstract 
Let $\mathcal{M}$ be a semifinite von Neumann algebra on a Hilbert space $\cH$ equipped with a faithful normal semifinite trace $\tau$, $S(\mathcal{M},\tau)$ be the ${}^*$algebra of all $\tau$measurable operators. Let $S_0(\mathcal{M},\tau)$ be the ${}^*$algebra of all $\tau$compact operators and
$T(\mathcal{M},\tau)=S_0(\mathcal{M},\tau)+\mathbb{C}I$ be the ${}^*$algebra of all operators $X=A+\lambda I$
with $A\in S_0(\mathcal{M},\tau)$ and $\lambda \in \mathbb{C}$. We prove that every operator of $T(\mathcal{M},\tau)$ that is leftinvertible in $T(\mathcal{M},\tau)$ is in fact invertible in $T(\mathcal{M},\tau)$.
It is a generalization of Sterling Berberian theorem (1982) on the subalgebra of thin operators in $\cB (\cH)$.
For the singular value function $\mu(t; Q)$ of $Q=Q^2\in S(\mathcal{M},\tau)$ we have $\mu(t; Q)\in \{0\}\bigcup
[1, +\infty)$ for all $t>0$. It gives the positive answer to the question posed by Daniyar Mushtari in 2010. 
ru_RU 
dc.language.iso 
ru 
ru_RU 
dc.subject 
Hilbert space 
ru_RU 
dc.subject 
von Neumann algebra 
ru_RU 
dc.subject 
semifinite trace 
ru_RU 
dc.subject 
$\tau$measurable operator 
ru_RU 
dc.subject 
$\tau$compact operator 
ru_RU 
dc.subject 
singular value function 
ru_RU 
dc.subject 
idempotent 
ru_RU 
dc.title 
The algebra of thin measurable operators is directly finite 
ru_RU 
dc.type 
Articles in international journals and collections 
ru_RU 
