Abstract
It is proved that if a normal semifinite weight ϕ on a von Neumann algebra \(\mathcal{M}\) satisfies the inequality \(\phi (|a_1 + a_2 |) \leqslant \phi (|a_1 |) + \phi (|a_2 |)\) for any selfadjoint operators \(a_1 ,a_2 \) in \(\mathcal{M}\), then this weight is a trace. Several similar characterizations of traces among the normal semifinite weights are proved. In particular, Gardner's result on the characterization of traces by the inequality \(|\phi (a)|{\text{ }} \leqslant {\text{ }}\phi (|a|)\) is refined and reinforced.
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Stolyarov, A.I., Tikhonov, O.E. & Sherstnev, A.N. Characterization of Normal Traces on Von Neumann Algebras by Inequalities for the Modulus. Mathematical Notes 72, 411–416 (2002). https://doi.org/10.1023/A:1020559623287
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DOI: https://doi.org/10.1023/A:1020559623287