KFU. Card publication. The algebra of thin measurable operators is directly finite

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FEDERAL UNIVERSITY

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THE ALGEBRA OF THIN MEASURABLE OPERATORS IS DIRECTLY FINITE

Form of presentation

Articles in international journals and collections

Year of publication

2023

Язык

английский

Authors, employees of KFU

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. 1--5.

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 left-invertible 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

On-line 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

File name	Size (MB)	Format
10.33205_cma.1181495_2676924.pdf	0,32	pdf	show / download

Full metadata record

Field DC	Value	Language
dc.contributor.author	Bikchentaev Ayrat Midkhatovich	ru_RU
dc.date.accessioned	2023-01-01T00:00:00Z	ru_RU
dc.date.available	2023-01-01T00:00:00Z	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. 1--5.	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 left-invertible 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