Unitarily invariant norms related to semi-finite factors

Junsheng Fang¹ ; Don Hadwin²

¹ School of Mathematical Sciences Dalian University of Technology Dalian, China, 116024
² Department of Mathematics University of New Hampshire Durham, NH 03824, U.S.A.

Studia Mathematica, Tome 229 (2015) no. 1, pp. 13-44

Résumé

Let $\mathcal M$ be a semi-finite factor and let $\mathcal J(\mathcal M)$ be the set of operators $T$ in $\mathcal M$ such that $T=ETE$ for some finite projection $E$. We obtain a representation theorem for unitarily invariant norms on $\mathcal J(\mathcal M)$ in terms of Ky Fan norms. As an application, we prove that the class of unitarily invariant norms on $\mathcal J(\mathcal M)$ coincides with the class of symmetric gauge norms on a classical abelian algebra, which generalizes von Neumann's classical 1940 result on unitarily invariant norms on $M_n(\mathbb C)$. As another application, Ky Fan's dominance theorem of 1951 is obtained for semi-finite factors.

Détail

DOI : 10.4064/sm8019-12-2015

Keywords: mathcal semi finite factor mathcal mathcal set operators mathcal ete finite projection obtain representation theorem unitarily invariant norms mathcal mathcal terms fan norms application prove class unitarily invariant norms mathcal mathcal coincides class symmetric gauge norms classical abelian algebra which generalizes von neumanns classical result unitarily invariant norms mathbb another application fans dominance theorem nbsp obtained semi finite factors