КФУ. Карточка публикации. The topologies of local convergence in measure on the algebra of measurable operators

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THE TOPOLOGIES OF LOCAL CONVERGENCE IN MEASURE ON THE ALGEBRA OF MEASURABLE OPERATORS

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

Bikchentaev A.M. The topologies of local convergence in measure on the algebra of measurable operators / A.M. Bikchentaev // Siberian Math. J. - 2023. - V. 64. - No 1. - P. 13--21.

Аннотация

Given a von Neumann algebra $M$ of operators on a Hilbert space $H$ and a faithful normal semifinite trace $\tau$ on $M$, denote by $S(M, \tau)$ the *-algebra of $\tau$-measurable operators. We obtain a sufficient condition for the positivity of an hermitian operator in $S(M, \tau)$ in terms of the topology $t_{\tau l}$ of $\tau$-local convergence in measure. We prove that the *-ideal $F(M, \tau)$ of elementary operators is $t_{\tau l}$-dense in $S(M, \tau)$. If $t_{\tau}$ is locally convex then so is $t_{\tau l}$; if $t_{\tau l}$ is locally convex then so is the topology $t_{w \tau l}$ of weakly $\tau$-local convergence in measure. We propose some method for constructing F-normed ideal spaces, henceforth F-NIPs, on $(M, \tau)$ starting from a prescribed F-NIP and preserving completeness, local convexity, local boundedness, or normability whenever present in the original. Given two F-NIPs $X$ and $Y$ on $(M, \tau)$, suppose that $AX\subseteq Y$ for some operator $A \in S(M, \tau)$. Then the multiplier $M_AX = AX$ acting as $M_A : X \to Y$ is continuous. In particular, for $X\subseteq Y$ the natural embedding of $X$ into $Y$ is continuous. We inspect the properties of decreasing sequences of F-NIPs on $(M, \tau)$.

Ключевые слова

Hilbert space, linear operator, von Neumann algebra, normal trace, measurable operator, local convergence in measure, locally convex space

Название журнала

SIBERIAN MATHEMATICAL JOURNAL

Ссылка для РПД

http://dspace.kpfu.ru/xmlui/bitstream/handle/net/173594/F_Print.pdf?sequence=1&isAllowed=y

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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	Bikchentaev A.M. The topologies of local convergence in measure on the algebra of measurable operators / A.M. Bikchentaev // Siberian Math. J. - 2023. - V. 64. - No 1. - P. 13--21.	ru_RU
dc.identifier.uri	https://repository.kpfu.ru/?p_id=275499	ru_RU
dc.description.abstract	SIBERIAN MATHEMATICAL JOURNAL	ru_RU
dc.description.abstract	Given a von Neumann algebra $M$ of operators on a Hilbert space $H$ and a faithful normal semifinite trace $\tau$ on $M$, denote by $S(M, \tau)$ the -algebra of $\tau$-measurable operators. We obtain a sufficient condition for the positivity of an hermitian operator in $S(M, \tau)$ in terms of the topology $t_{\tau l}$ of $\tau$-local convergence in measure. We prove that the -ideal $F(M, \tau)$ of elementary operators is $t_{\tau l}$-dense in $S(M, \tau)$. If $t_{\tau}$ is locally convex then so is $t_{\tau l}$; if $t_{\tau l}$ is locally convex then so is the topology $t_{w \tau l}$ of weakly $\tau$-local convergence in measure. We propose some method for constructing F-normed ideal spaces, henceforth F-NIPs, on $(M, \tau)$ starting from a prescribed F-NIP and preserving completeness, local convexity, local boundedness, or normability whenever present in the original. Given two F-NIPs $X$ and $Y$ on $(M, \tau)$, suppose that $AX\subseteq Y$ for some operator $A \in S(M, \tau)$. Then the multiplier $M_AX = AX$ acting as $M_A : X \to Y$ is continuous. In particular, for $X\subseteq Y$ the natural embedding of $X$ into $Y$ is continuous. We inspect the properties of decreasing sequences of F-NIPs on $(M, \tau)$.	ru_RU
dc.language.iso	ru	ru_RU
dc.subject	Hilbert space	ru_RU
dc.subject	linear operator	ru_RU
dc.subject	von Neumann algebra	ru_RU
dc.subject	normal trace	ru_RU
dc.subject	measurable operator	ru_RU
dc.subject	local convergence in measure	ru_RU
dc.subject	locally convex space	ru_RU
dc.title	The topologies of local convergence in measure on the algebra of measurable operators	ru_RU
dc.type	Статьи в зарубежных журналах и сборниках	ru_RU