КФУ. Карточка публикации. The algebra of thin measurable operators is directly finite

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THE ALGEBRA OF THIN MEASURABLE OPERATORS IS DIRECTLY FINITE

Форма представления

Статьи в зарубежных журналах и сборниках

Год публикации

2023

Язык

английский

Авторы, сотрудники КФУ

Бикчентаев Айрат Мидхатович, автор

Библиографическое описание на языке оригинала

Airat M. Bikchentaev, The algebra of thin measurable operators is directly finite // Constructive Mathematical Analysis. 2023. V. 6, no 1. P. 1--5.

Аннотация

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.

Ключевые слова

Hilbert space, von Neumann algebra, semifinite trace, $\tau$-measurable operator, $\tau$-compact operator, singular value function, idempotent

Название журнала

Constructive Mathematical Analysis

Ссылка для РПД

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

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https://repository.kpfu.ru/?p_id=276828

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Полная запись метаданных

Поле DC	Значение	Язык
dc.contributor.author	Бикчентаев Айрат Мидхатович	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/?p_id=276828	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	Статьи в зарубежных журналах и сборниках	ru_RU