KFU. Card publication. Projections and Traces on von Neumann Algebras

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PROJECTIONS AND TRACES ON VON NEUMANN ALGEBRAS

Form of presentation

Articles in international journals and collections

Year of publication

2019

Язык

английский

Authors, employees of KFU

Bikchentaev Ayrat Midkhatovich, author

Authors, postgraduates KFU

Abed Sami Abdulla Abed, postgraduate kfu

Bibliographic description in the original language

Bikchentaev A.M., Abed S.A. Projections and Traces on von Neumann Algebras / A.M. Bikchentaev, S.A. Abed // Lobachevskii Journal of Mathematics. - 2019. - Vol. 40, No. 9. - P. 1260–1267.

Annotation

Let P,Q be projections on a Hilbert space. We prove the equivalence of the following conditions: (i) PQ + QP ≤ 2(QPQ)^p for some number 0 < p ≤ 1; (ii) PQ is paranormal; (iii) PQ is M*-paranormal; (iv) PQ = QP. This allows us to obtain the commutativity criterion for a von Neumann algebra. For a positive normal functional ϕ on von Neumann algebra M it is proved the equivalence of the following conditions: (i) ϕ is tracial; (ii) ϕ(PQ + QP) ≤ 2ϕ((QPQ)^p) for all projections P,Q∈M and for some p = p(P,Q) ∈ (0, 1]; (iii) ϕ(PQP) ≤ ϕ(P)^{1/p}ϕ(Q)^{1/q} for all projections P,Q∈M and some positive numbers p = p(P,Q), q = q(P,Q) with 1/p+ 1/q = 1, p = 2. Corollary: for a positive normal functional ϕ onMthe following conditions are equivalent: (i) ϕ is tracial; (ii) ϕ(A + A*) ≤ 2ϕ(|A*|) for all A∈M.

Keywords

Hilbert space, linear operator, projection, von Neumann algebra, positive functional, trace, operator inequality, commutativity

The name of the journal

Lobashevskii Journal of Mathematics

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

https://repository.kpfu.ru/eng/?p_id=210046&p_lang=2

Resource files

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LOJM1260.pdf	0,51	pdf	show / download

Full metadata record

Field DC	Value	Language
dc.contributor.author	Bikchentaev Ayrat Midkhatovich	ru_RU
dc.contributor.author	Abed Sami Abdulla Abed	ru_RU
dc.date.accessioned	2019-01-01T00:00:00Z	ru_RU
dc.date.available	2019-01-01T00:00:00Z	ru_RU
dc.date.issued	2019	ru_RU
dc.identifier.citation	Bikchentaev A.M., Abed S.A. Projections and Traces on von Neumann Algebras / A.M. Bikchentaev, S.A. Abed // Lobachevskii Journal of Mathematics. - 2019. - Vol. 40, No. 9. - P. 1260–1267.	ru_RU
dc.identifier.uri	https://repository.kpfu.ru/eng/?p_id=210046&p_lang=2	ru_RU
dc.description.abstract	Lobashevskii Journal of Mathematics	ru_RU
dc.description.abstract	Let P,Q be projections on a Hilbert space. We prove the equivalence of the following conditions: (i) PQ + QP ≤ 2(QPQ)^p for some number 0 < p ≤ 1; (ii) PQ is paranormal; (iii) PQ is M-paranormal; (iv) PQ = QP. This allows us to obtain the commutativity criterion for a von Neumann algebra. For a positive normal functional ϕ on von Neumann algebra M it is proved the equivalence of the following conditions: (i) ϕ is tracial; (ii) ϕ(PQ + QP) ≤ 2ϕ((QPQ)^p) for all projections P,Q∈M and for some p = p(P,Q) ∈ (0, 1]; (iii) ϕ(PQP) ≤ ϕ(P)^{1/p}ϕ(Q)^{1/q} for all projections P,Q∈M and some positive numbers p = p(P,Q), q = q(P,Q) with 1/p+ 1/q = 1, p = 2. Corollary: for a positive normal functional ϕ onMthe following conditions are equivalent: (i) ϕ is tracial; (ii) ϕ(A + A) ≤ 2ϕ(\|A*\|) for all A∈M.	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	positive functional	ru_RU
dc.subject	trace	ru_RU
dc.subject	operator inequality	ru_RU
dc.subject	commutativity	ru_RU
dc.title	Projections and Traces on von Neumann Algebras	ru_RU
dc.type	Articles in international journals and collections	ru_RU