KFU. Card publication. Trace inequalities and characterizations of tracial functionals in operator algebras

KAZAN
FEDERAL UNIVERSITY

- RUS
- 中文
- ES

TRACE INEQUALITIES AND CHARACTERIZATIONS OF TRACIAL FUNCTIONALS IN OPERATOR ALGEBRAS

Form of presentation

Articles in international journals and collections

Year of publication

2026

��

��

Authors, employees of KFU

Bikchentaev Ayrat Midkhatovich, author

Other authors

Moslehian Mohammad Sal , author

Bibliographic description in the original language

Airat M. Bikchentaev, Mohammad Sal Moslehian, Trace inequalities and characterizations of tracial functionals in operator algebras // Positivity (2026) V. 30, Article 24. 15 p.

Annotation

For a positive normal linear functional ϕ on a von Neumann algebra A , we prove that the following conditions are equivalent: (i) ϕ is tracial, (ii) |ϕ(Re(A2)| ≤ ϕ(|A|2) for all A ∈ A , and (iii) |ϕ(A2)| ≤ ϕ(|A|2) for all A ∈ A . Based on this result, we present some criteria for commutativity of a von Neumann algebra. For a trace ϕ on a C∗-algebra A , we prove that −ϕ(A2B2) ≤ ϕ((AB)2) ≤ ϕ(A2B2) for certain elements of A , and show that when ϕ is faithful, the equality in the second inequality is achieved if and only if AB = BA. Moreover, we partially generalize the Araki–Lieb–Thirring inequality to arbitrary traces on any C∗-algebras and to self-adjoint elements. Furthermore, we present a simple joint proof for Tr(AB) ? Tr(X∗X) ≤ Tr(A) Tr(B) ? Tr(X∗) Tr(X) provided that A XX∗ B is positive semidefinite, without using the fact that (X) = X + (Tr X)I is completely copositive, and then present a characterization of the trace on the full matrix algebra Mn.

Keywords

C∗-algebra, von Neumann algebra, trace, positive linear functional, positive semidefinite block matrix

The name of the journal

POSITIVITY

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

https://repository.kpfu.ru/eng/?p_id=324508&p_lang=2

Resource files

File name	Size (MB)	Format
F_s11117_026_01180_0.pdf	0,26	pdf	show / download

Full metadata record

Field DC	Value	Language
dc.contributor.author	Bikchentaev Ayrat Midkhatovich	ru_RU
dc.contributor.author	Moslehian Mohammad Sal	ru_RU
dc.date.accessioned	2026-01-01T00:00:00Z	ru_RU
dc.date.available	2026-01-01T00:00:00Z	ru_RU
dc.date.issued	2026	ru_RU
dc.identifier.citation	Airat M. Bikchentaev, Mohammad Sal Moslehian, Trace inequalities and characterizations of tracial functionals in operator algebras // Positivity (2026) V. 30, Article 24. 15 p.	ru_RU
dc.identifier.uri	https://repository.kpfu.ru/eng/?p_id=324508&p_lang=2	ru_RU
dc.description.abstract	POSITIVITY	ru_RU
dc.description.abstract	For a positive normal linear functional ϕ on a von Neumann algebra A , we prove that the following conditions are equivalent: (i) ϕ is tracial, (ii) \|ϕ(Re(A2)\| ≤ ϕ(\|A\|2) for all A ∈ A , and (iii) \|ϕ(A2)\| ≤ ϕ(\|A\|2) for all A ∈ A . Based on this result, we present some criteria for commutativity of a von Neumann algebra. For a trace ϕ on a C∗-algebra A , we prove that −ϕ(A2B2) ≤ ϕ((AB)2) ≤ ϕ(A2B2) for certain elements of A , and show that when ϕ is faithful, the equality in the second inequality is achieved if and only if AB = BA. Moreover, we partially generalize the Araki–Lieb–Thirring inequality to arbitrary traces on any C∗-algebras and to self-adjoint elements. Furthermore, we present a simple joint proof for Tr(AB) ? Tr(X∗X) ≤ Tr(A) Tr(B) ? Tr(X∗) Tr(X) provided that A XX∗ B is positive semidefinite, without using the fact that (X) = X + (Tr X)I is completely copositive, and then present a characterization of the trace on the full matrix algebra Mn.	ru_RU
dc.language.iso	ru	ru_RU
dc.subject	C∗-algebra	ru_RU
dc.subject	von Neumann algebra	ru_RU
dc.subject	trace	ru_RU
dc.subject	positive linear functional	ru_RU
dc.subject	positive semidefinite block matrix	ru_RU
dc.title	Trace inequalities and characterizations of tracial functionals in operator algebras	ru_RU
dc.type	Articles in international journals and collections	ru_RU