Kazan (Volga region) Federal University, KFU
Form of presentationArticles in international journals and collections
Year of publication2019
  • Bikchentaev Ayrat Midkhatovich, author
  • 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
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