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The topologies of local convergence in measure on the algebra of measurable operators

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dc.contributor Казанский федеральный университет
dc.contributor.author Bikchentaev Airat Midkhatovich
dc.date.accessioned 2023-01-19T07:49:29Z
dc.date.available 2023-01-19T07:49:29Z
dc.date.issued 2023
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.
dc.identifier.uri https://dspace.kpfu.ru/xmlui/handle/net/173594
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 $(M, \tau)$.
dc.language.iso en
dc.relation.ispartofseries SIBERIAN MATHEMATICAL JOURNAL
dc.rights открытый доступ
dc.subject Hilbert space
dc.subject linear operator
dc.subject von Neumann algebra
dc.subject normal trace
dc.subject measurable operator
dc.subject local convergence in measure
dc.subject locally convex space
dc.subject.other Математика
dc.title The topologies of local convergence in measure on the algebra of measurable operators
dc.type Article
dc.contributor.org Институт математики и механики им. Н.И. Лобачевского
dc.description.pages 13-21
dc.relation.ispartofseries-issue 1
dc.relation.ispartofseries-volume 64
dc.pub-id 275499


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