| Форма представления | Статьи в зарубежных журналах и сборниках |
| Год публикации | 2021 |
| Язык | английский |
|
Бикчентаев Айрат Мидхатович, автор
Иваньшин Петр Николаевич, автор
|
| Библиографическое описание на языке оригинала |
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. |
| Аннотация |
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. |
| Ключевые слова |
Hilbert space, linear operator,
projection, von Neumann algebra, tracial state, independence. |
| Название журнала |
Lobachevskii Journal of Mathematics
|
| Пожалуйста, используйте этот идентификатор, чтобы цитировать или ссылаться на эту карточку |
https://repository.kpfu.ru/?p_id=255107 |
| Файлы ресурса | |
|
|
Полная запись метаданных  |
| Поле DC |
Значение |
Язык |
| dc.contributor.author |
Бикчентаев Айрат Мидхатович |
ru_RU |
| dc.contributor.author |
Иваньшин Петр Николаевич |
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/?p_id=255107 |
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 |
Статьи в зарубежных журналах и сборниках |
ru_RU |
|
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