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 $(M, \tau)$. |
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dc.language.iso |
en |
|
dc.relation.ispartofseries |
SIBERIAN MATHEMATICAL JOURNAL |
|
dc.rights |
открытый доступ |
|
dc.subject |
Hilbert space |
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dc.subject |
linear operator |
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dc.subject |
von Neumann algebra |
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dc.subject |
normal trace |
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dc.subject |
measurable operator |
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dc.subject |
local convergence in measure |
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dc.subject |
locally convex space |
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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 |
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dc.relation.ispartofseries-issue |
1 |
|
dc.relation.ispartofseries-volume |
64 |
|
dc.pub-id |
275499 |
|