Form of presentation  Articles in international journals and collections 
Year of publication  2023 
Язык  английский 

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. 1321. 
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
Fnormed ideal spaces, henceforth FNIPs, on $(M, \tau)$ starting from a prescribed FNIP and preserving
completeness, local convexity, local boundedness, or normability whenever present in the original. Given
two FNIPs $X$ 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 FNIPs 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

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

Please use this ID to quote from or refer to the card 
https://repository.kpfu.ru/eng/?p_id=275499&p_lang=2 
Resource files  

Full metadata record 
Field DC 
Value 
Language 
dc.contributor.author 
Bikchentaev Ayrat Midkhatovich 
ru_RU 
dc.date.accessioned 
20230101T00:00:00Z 
ru_RU 
dc.date.available 
20230101T00:00:00Z 
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. 1321. 
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
Fnormed ideal spaces, henceforth FNIPs, on $(M, \tau)$ starting from a prescribed FNIP and preserving
completeness, local convexity, local boundedness, or normability whenever present in the original. Given
two FNIPs $X$ 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 FNIPs 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 
