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Characterization of Tracial Functionals on Von Neumann Algebras

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Abstract

It is proved that the inequality

$$\varphi(A)\leq\varphi(|A+\textrm{i}B|)\quad\text{for all}\quad A\in\mathcal{A}^{+}\quad\text{and}\quad B\in\mathcal{A}^{\text{sa}}$$

characterizes tracial functionals among all positive normal functionals \(\varphi\) on a von Neumann algebra \(\mathcal{A}\). This strengthens the L. T. Gardner’s characterization (1979). As a consequence, a criterion for commutativity of von Neumann algebras is obtained. Also we give a characterization of traces in a wide class of weights on a von Neumann algebra via this inequality. Every faithful normal semifinite trace \(\varphi\) on a von Neumann algebra \(\mathcal{A}\) satisfies this relation. Let \(|||\cdot|||\) be a unitarily invariant norm on a unital \(C^{*}\)-algebra \(\mathcal{A}\). Then \(|||A|||\leq|||A+\textrm{i}B|||\) for all \(A\in\mathcal{A}^{+}\) and \(B\in\mathcal{A}^{\text{sa}}\).

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Funding

The work was carried out as part of the development program of the Scientific and Educational Mathematical Center of the Volga Federal District, agreement no. 075-02-2020-1478.

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Authors and Affiliations

  1. N. I. Lobachevskii Institute of Mathematics and Mechanics, Kazan (Volga Region) Federal University, 420008, Kazan, Tatarstan, Russia

    Hasan Alhasan & Khattab Fawwaz

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  1. Hasan Alhasan
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  2. Khattab Fawwaz
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Correspondence to Hasan Alhasan or Khattab Fawwaz.

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(Submitted by G. G. Amosov)

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Alhasan, H., Fawwaz, K. Characterization of Tracial Functionals on Von Neumann Algebras. Lobachevskii J Math 42, 2273–2279 (2021). https://doi.org/10.1134/S1995080221100024

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  • DOI: https://doi.org/10.1134/S1995080221100024

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