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Let M be a von Neumann algebra and let (p and y be bounded linear functionals on M. We then have the norm inequality l^ + Vll^ WI + lVl l On the other hand, it is well-known that if (p and if/ are positive, then ||<p + y|| HW + MIIn general, however, such an equality does not necessarily holds if both <p and iff are not positive. The purpose of this paper is to investigate when the norm equality ||9 + vll HMI + W holds for given normal linear functionals <p and y. Then the fact to play an essential role is the following: Let M be a von Neumann algebra and let (p be a normal linear functional on M. Then we have

Norm additivity conditions for normal linear functionals on von Neumann algebras