Norm additivity conditions for normal linear functionals on von Neumann algebras (Q1913413)

Let \(M\) be a von Neumann algebra and let \(\varphi\) and \(\psi\) be bounded linear functionals on \(M\). We then have the norm inequality \(|\varphi+ \psi|\leq |\varphi|+ |\psi|\). On the other hand, it is well-known that if \(\varphi\) and \(\psi\) are positive, then \(|\varphi+ \psi|= |\varphi|+ |\psi|\). In general, however, such an equality does not necessarily hold if both \(\varphi\) and \(\psi\) are not positive. The purpose of this paper is to investigate when the norm equality \(|\varphi+ \psi|= |\varphi|+ |\psi|\) holds for given normal linear functionals \(\varphi\) and \(\psi\). Then the fact to play an essential role is the following: Let \(M\) be a von Neumann algebra and let \(\varphi\) be a normal linear functional on \(M\). Then we have \[ \varphi(\cdot)= |\varphi |(\nu\cdot),\;|\varphi|(\cdot)= \varphi(\nu^*\cdot),\text{ and } |\varphi|= |\;|\varphi|\;| \] for all partial isometries \(\nu^*\) in \(M\) satisfying that \(\varphi(\nu^*)= |\varphi|\), where \(|\varphi|\) denotes the absolute value of \(\varphi\). In connection with this fact, we may expect that the set of those elements \(x\), in the unit ball of \(M\), with \(\varphi(x)= |\varphi |\) has nice information on norms and absolute values of normal linear functionals on \(M\). In fact, by employing such a set, we will give necessary and sufficient conditions for a pair of normal linear functionals \(\varphi\) and \(\psi\) to satisfy that \(|\varphi+ \psi|= |\varphi|+ |\psi|\) or that \(|\varphi+ \psi|= |\varphi|+ |\psi|\).