Characterization of normal traces on von Neumann algebras by inequalities for the modulus

It is proved that if a normal semifinite weight phi on a von Neumann algebra M satisfies the inequality phi(\a(1) + a(2)\) less than or equal to phi(\a(1)\) + phi(\a(2)\) for any selfadjoint operators a(1), a(2) in 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)\ less than or equal to phi(\a\) is refined and reinforced.