KFU. Card publication. Characterization of certain traces on von Neumann algebras

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CHARACTERIZATION OF CERTAIN TRACES ON VON NEUMANN ALGEBRAS

Form of presentation

Articles in international journals and collections

Year of publication

2022

Язык

английский

Authors, employees of KFU

Bikchentaev Ayrat Midkhatovich, author

Bibliographic description in the original language

A. Bikchentaev, Characterization of certain traces on von Neumann algebras // L. Accardi et al. (eds.), Infinite Dimensional Analysis, Quantum Probability and Applications. ICQPRT 2021 (Springer Proceedings in Mathematics & Statistics 390): QP41 Conference, Al Ain, UAE, March 28–April 1, 2021, Springer, 2022. -- P. 279--289.

Annotation

L. Accardi et al. (eds.), Infinite Dimensional Analysis, Quantum Probability and Applications. ICQPRT 2021 (Springer Proceedings in Mathematics & Statistics 390): QP41 Conference, Al Ain, UAE, March 28–April 1, 2021

Keywords

Hilbert space, linear operator, von Neumann algebra, $C^*$-algebra, block projection operator, weight, trace, tracial inequality, commutativity

The name of the journal

On-line resource for training course

http://dspace.kpfu.ru/xmlui/bitstream/handle/net/173179/bik_proc2.pdf?sequence=1&isAllowed=y

Please use this ID to quote from or refer to the card

https://repository.kpfu.ru/eng/?p_id=271347&p_lang=2

Resource files

File name	Size (MB)	Format
bik_proc2.pdf	0,33	pdf	show / download

Full metadata record

Field DC	Value	Language
dc.contributor.author	Bikchentaev Ayrat Midkhatovich	ru_RU
dc.date.accessioned	2022-01-01T00:00:00Z	ru_RU
dc.date.available	2022-01-01T00:00:00Z	ru_RU
dc.date.issued	2022	ru_RU
dc.identifier.citation	A. Bikchentaev, Characterization of certain traces on von Neumann algebras // L. Accardi et al. (eds.), Infinite Dimensional Analysis, Quantum Probability and Applications. ICQPRT 2021 (Springer Proceedings in Mathematics & Statistics 390): QP41 Conference, Al Ain, UAE, March 28–April 1, 2021, Springer, 2022. -- P. 279--289.	ru_RU
dc.identifier.uri	https://repository.kpfu.ru/eng/?p_id=271347&p_lang=2	ru_RU
dc.description.abstract	L. Accardi et al. (eds.), Infinite Dimensional Analysis, Quantum Probability and Applications. ICQPRT 2021 (Springer Proceedings in Mathematics & Statistics 390): QP41 Conference, Al Ain, UAE, March 28–April 1, 2021	ru_RU
dc.description.abstract	Consider a unital $C^$-algebra $\mathcal{A}$. Let $n\geq 2$ and let $P_1, \ldots , P_n$ be projections in $\mathcal{A}$ such that $P_1 + \ldots +P_n=I$. We costruct $\mathcal{P}_n\colon \mathcal{A}\to \mathcal{A}$ being a block projection operator given by the formula $\mathcal{P}_n(X)=\sum_{k=1}^n P_kXP_k$ for all $X\in \mathcal{A}$. For a weight $\varphi$ on a von Neumann algebra $\mathcal{A}$, we prove that $\varphi$ is a trace if and only if $\varphi (\mathcal{P}_2(A))=\varphi (A)$ for all $A\in \mathcal{A}^+$. We also prove that if $\mathcal{A}$ is a von Neumann algebra then for a normal semifinite weight $\varphi$ on $\mathcal{A}$ the following conditions are equivalent: {\rm (i)} $\varphi$ is a trace; {\rm (ii)} $\varphi((A^{m/2}B^mA^{m/2} )^k)\leq\varphi ((A^{k/2}B^kA^{k/2})^m)$ for all $A, B\in\mathcal{A}^+$ and some numbers $k,m \in\mathbb{R}$ such that $k>m>0$; {\rm (iii)} $\varphi (\|\mathcal{P}_n(A)\|)\leq\varphi (\|A\|)$ for all $A\in \mathcal{A}$ and for all projections $P_1, \ldots , P_n\in \mathcal{A}$. As a consequence, we obtain a criterions for commutativity of von Neumann algebras and $C^$-algebras.	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	$C^*$-algebra	ru_RU
dc.subject	block projection operator	ru_RU
dc.subject	weight	ru_RU
dc.subject	trace	ru_RU
dc.subject	tracial inequality	ru_RU
dc.subject	commutativity	ru_RU
dc.title	Characterization of certain traces on von Neumann algebras	ru_RU
dc.type	Articles in international journals and collections	ru_RU