KFU. Card publication. On Independence of Events in Noncommutative Probability Theory

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ON INDEPENDENCE OF EVENTS IN NONCOMMUTATIVE PROBABILITY THEORY

Form of presentation

Articles in international journals and collections

Year of publication

2021

Язык

английский

Authors, employees of KFU

Bikchentaev Ayrat Midkhatovich, author

Ivanshin Petr Nikolaevich, author

Bibliographic description in the original language

A. M. Bikchentaev, P. N. Ivanshin, On Independence of Events in Noncommutative Probability Theory // Lobachevskii Journal of Mathematics, 2021, Vol. 42, No. 10, pp. 2306–2314.

Annotation

We consider a tracial state $\varphi$ on a von Neumann algebra $\mathcal{A}$ and assume that projections $P, Q$ of $\mathcal{A}$ are independent if $\varphi (PQ)=\varphi (P)\varphi (Q)$. First we present the general criterion of a projection pair independence. We then give a geometric criterion for independence of different pairs of projections. If atoms $P$ and $Q$ are independent then $\varphi (P)= \varphi (Q)$. Also here we deal with an analog of a ``symmetric difference'' for a pair of projections $P$ and $ Q$, namely, the projection $ R\equiv P\vee Q -P\wedge Q$. If $R\neq 0, I$, the pairs $\{P, R\}$ and $ \{Q, R\}$ are independent then $\varphi (P)= \varphi (Q)=1/2$ and $\varphi ( P\wedge Q + P\vee Q) =1$. If, moreover, $P$ and $ Q $ are independent, then $\varphi ( P\wedge Q)\leq 1/4$ and $\varphi ( P\vee Q)\geq 3/4$. For an atomless von Neumann algebra $\mathcal{A}$ a tracial normal state is determined by its specification of independent events. We clarify our considerations with examples of projection pairs with the differemt mutual independency relations. For the full matrix algebra we give several equivalent conditions for the independence of pairs of projections.

Keywords

Hilbert space, linear operator, projection, von Neumann algebra, tracial state, independence.

The name of the journal

Lobachevskii Journal of Mathematics

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

https://repository.kpfu.ru/eng/?p_id=255107&p_lang=2

Resource files

File name	Size (MB)	Format
LobJMat2110006BikchentaevKOR.pdf	0,48	pdf	show / download

Full metadata record

Field DC	Value	Language
dc.contributor.author	Bikchentaev Ayrat Midkhatovich	ru_RU
dc.contributor.author	Ivanshin Petr Nikolaevich	ru_RU
dc.date.accessioned	2021-01-01T00:00:00Z	ru_RU
dc.date.available	2021-01-01T00:00:00Z	ru_RU
dc.date.issued	2021	ru_RU
dc.identifier.citation	A. M. Bikchentaev, P. N. Ivanshin, On Independence of Events in Noncommutative Probability Theory // Lobachevskii Journal of Mathematics, 2021, Vol. 42, No. 10, pp. 2306–2314.	ru_RU
dc.identifier.uri	https://repository.kpfu.ru/eng/?p_id=255107&p_lang=2	ru_RU
dc.description.abstract	Lobachevskii Journal of Mathematics	ru_RU
dc.description.abstract	We consider a tracial state $\varphi$ on a von Neumann algebra $\mathcal{A}$ and assume that projections $P, Q$ of $\mathcal{A}$ are independent if $\varphi (PQ)=\varphi (P)\varphi (Q)$. First we present the general criterion of a projection pair independence. We then give a geometric criterion for independence of different pairs of projections. If atoms $P$ and $Q$ are independent then $\varphi (P)= \varphi (Q)$. Also here we deal with an analog of a ``symmetric difference'' for a pair of projections $P$ and $ Q$, namely, the projection $ R\equiv P\vee Q -P\wedge Q$. If $R\neq 0, I$, the pairs $\{P, R\}$ and $ \{Q, R\}$ are independent then $\varphi (P)= \varphi (Q)=1/2$ and $\varphi ( P\wedge Q + P\vee Q) =1$. If, moreover, $P$ and $ Q $ are independent, then $\varphi ( P\wedge Q)\leq 1/4$ and $\varphi ( P\vee Q)\geq 3/4$. For an atomless von Neumann algebra $\mathcal{A}$ a tracial normal state is determined by its specification of independent events. We clarify our considerations with examples of projection pairs with the differemt mutual independency relations. For the full matrix algebra we give several equivalent conditions for the independence of pairs of projections.	ru_RU
dc.language.iso	ru	ru_RU
dc.subject	Hilbert space	ru_RU
dc.subject	linear operator	ru_RU
dc.subject	projection	ru_RU
dc.subject	von Neumann algebra	ru_RU
dc.subject	tracial state	ru_RU
dc.subject	independence.	ru_RU
dc.title	On Independence of Events in Noncommutative Probability Theory	ru_RU
dc.type	Articles in international journals and collections	ru_RU