Subharmonicity in von Neumann algebras
Volume 170 / 2005
Studia Mathematica 170 (2005), 219-226
MSC: Primary 46L10; Secondary 15A18, 31A05.
DOI: 10.4064/sm170-3-1
Abstract
Let ${\cal M}$ be a von Neumann algebra with unit $1_{\cal M}$. Let $\tau$ be a faithful, normal, semifinite trace on ${\cal M}$. Given $x\in{\cal M}$, denote by $\mu_t(x)_{t\ge0}$ the generalized $s$-numbers of $x$, defined by $$ \mu_t(x)=\inf\{\|xe\|: e \hbox{ is a projection in ${\cal M}$ with }\tau(1_{\cal M}-e)\le t\} \quad (t\ge0). $$ We prove that, if $D$ is a complex domain and $f:D\to{\cal M}$ is a holomorphic function, then, for each $t\ge0$, $\lambda\mapsto\int_0^t\log\mu_s(f(\lambda))\,ds$ is a subharmonic function on $D$. This generalizes earlier subharmonicity results of White and Aupetit on the singular values of matrices.
