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Metrics on Projections of the Von Neumann Algebra Associated with Tracial Functionals

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Abstract

Let φ be a positive functional on a von Neumann algebra \(\mathscr{A}\) and let \(\mathscr{A}^{\rm{pr}}\) be the projection lattice in \(\mathscr{A}\). Given \(P,Q \in \mathscr{A}^{\rm{pr}}\), put ρφ(P, Q) = φ(∣P − Q∣) and dφ(P, Q) = φ(P ∨ Q − P ∧ Q). Then ρφ(P, Q) ≤ dφ(P, Q) and ρφ(P, Q) = dφ(P, Q) provided that PQ = QP. The mapping ρφ (or dφ) meets the triangle inequality if and only if φ is a tracial functional. If τ is a faithful tracial functional then ρτ and dτ are metrics on \(\mathscr{A}^{\rm{pr}}\). Moreover, if τ is normal then (\(\mathscr{A}^{\rm{pr}}\), ρτ) and (\(\mathscr{A}^{\rm{pr}}\), dτ) are complete metric spaces. Convergences with respect to ρτ and dτ are equivalent if and only if \(\mathscr{A}\) is abelian; in this case ρτ = dτ. We give one more criterion for commutativity of \(\mathscr{A}\) in terms of inequalities.

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Authors and Affiliations

  1. Lobachevskii institute of mathematics and mechanics of Kazan (Volga Region), Federal University, Kazan, Russia

    A. M. Bikchentaev

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  1. A. M. Bikchentaev
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Correspondence to A. M. Bikchentaev.

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The author is grateful to the referee for valuable advice.

Russian Text © The Author(s), 2019, published in Sibirskii Matematicheskii Zhurnal, 2019, Vol. 60, No. 6, pp. 1223–1228.

The research was funded by the subsidy allocated to Kazan Federal University for the state assignment in the sphere of scientific activities, project 1.9773.2017/8.9.

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Bikchentaev, A.M. Metrics on Projections of the Von Neumann Algebra Associated with Tracial Functionals. Sib Math J 60, 952–956 (2019). https://doi.org/10.1134/S003744661906003X

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  • DOI: https://doi.org/10.1134/S003744661906003X

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