KFU. Card publication. 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

Form of presentation

Articles in international journals and collections

Year of publication

2023

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Authors, employees of KFU

Bikchentaev Ayrat Midkhatovich, author

Bibliographic description in the original language

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.

Annotation

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 $(M, \tau)$.

Keywords

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

The name of the journal

SIBERIAN MATHEMATICAL JOURNAL

On-line resource for training course

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

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https://repository.kpfu.ru/eng/?p_id=275499&p_lang=2

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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	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/eng/?p_id=275499&p_lang=2	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	Articles in international journals and collections	ru_RU