Subharmonicity in von Neumann algebras

Thomas Ransford¹ ; Michel Valley²

¹Département de mathématiques et de statistique Université Laval Québec (QC), Canada G1K 7P4
²Département de mathématiques et de statistique Université Laval Québec (QC) Canada G1K 7P4

Studia Mathematica, Tome 170 (2005) no. 3, pp. 219-226

Résumé

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.

Détail

DOI : 10.4064/sm170-3-1

Keywords: cal von neumann algebra unit cal tau faithful normal semifinite trace cal given cal denote generalized s numbers defined inf hbox projection cal tau cal e quad prove complex domain cal holomorphic function each lambda mapsto int log lambda subharmonic function generalizes earlier subharmonicity results white aupetit singular values matrices