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Characterization of certain traces on von Neumann algebras

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dc.contributor Казанский федеральный университет
dc.contributor.author Bikchentaev Airat Midkhatovich
dc.date.accessioned 2022-10-10T08:07:44Z
dc.date.available 2022-10-10T08:07:44Z
dc.date.issued 2022
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.
dc.identifier.uri https://dspace.kpfu.ru/xmlui/handle/net/173179
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.
dc.language.iso en
dc.relation.ispartofseries 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
dc.rights открытый доступ
dc.subject Hilbert space
dc.subject linear operator
dc.subject von Neumann algebra
dc.subject $C^*$-algebra
dc.subject block projection operator
dc.subject weight
dc.subject trace
dc.subject tracial inequality
dc.subject commutativity
dc.subject.other Математика
dc.title Characterization of certain traces on von Neumann algebras
dc.type Article
dc.contributor.org Институт математики и механики им. Н.И. Лобачевского
dc.description.pages 279-289
dc.relation.ispartofseries-volume 390
dc.pub-id 271347


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