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Characterization of Normal Traces on Von Neumann Algebras by Inequalities for the Modulus

Authors:	Stolyarov A I Tikhonov O E Sherstnev A N

Institution:	(1) Kazan State University, Russia

Abstract:	It is proved that if a normal semifinite weight 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 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.

Keywords:	von Neumann algebra normal semifinite weight trace ultrastrong topology ultraweak topology
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